SUPREMUM, INFIMUM AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF COLOMBEAU GENERALIZED NUMBERS
A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO
Abstract. It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers Re does not generalize classical 1 results. E.g. the sequence n 6→ 0 and a sequence (xn)n∈N converges if and only if xn+1 − xn → 0. This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that Re is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non- Archimedean settings.
1. Introduction A key concept of non-Archimedean analysis is that extending the real field R into a ring containing infinitesimals and infinite numbers could eventually lead to the solution of non trivial problems. This is the case, e.g., of Colombeau theory, where nonlinear generalized functions can be viewed as set-theoretical maps on domains consisting of generalized points of the non-Archimedean ring Re. This orientation has become increasingly important in recent years and hence it has led to the study of preliminary notions of Re (cf., e.g., [17,3,1, 18,2,5, 26, 16, 12]; see below for a self-contained introduction to the ring of Colombeau generalized numbers Re). In particular, the sharp topology on Re (cf., e.g., [10, 22, 23] and below) is the appropriate notion to deal with continuity of this class of generalized functions and for a suitable concept of well-posedness. This topology necessarily has to deal with 1 balls having infinitesimal radius r ∈ Re, and thus n 6→ 0 if n → +∞, n ∈ N, because 1 we never have R>0 3 n < r if r is infinitesimal. Another unusual property related to the sharp topology can be derived from the following inequalities (where m ∈ N, 2 n ∈ N≤m, r ∈ Re>0 is an infinitesimal number, and |xk+1 − xk| ≤ r ) 2 |xm − xn| ≤ |xm − xm−1| + ... + |xn+1 − xn| ≤ (m − n)r < r,
2000 Mathematics Subject Classification. 46F-XX, 46F30, 26E30. Key words and phrases. Colombeau generalized numbers, non-Archimedean rings, generalized functions. A. Mukhammadiev has been supported by grant P 30407 of the Austrian Science Fund FWF.. D. Tiwari has been supported by grant P 30407 of the Austrian Science Fund FWF.. P. Giordano has been supported by grants P30407, P25311 and P25116 of the Austrian Science Fund FWF. 1 2 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO
N which imply that (xn)n∈N ∈ Re is a Cauchy sequence if and only if |xn+1 − xn| → 0 (actually, this is a well-known property of every ultrametric space, cf., e.g., [15, 22]). Naturally, this has several counter-intuitive consequences (arising from differences with the classical theory) when we have to deal with the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions (cf., e.g., [20, 25, 13]). One of the aims of the present article is to solve this kind of counter-intuitive properties so as to arrive at useful notions for the theory of generalized functions. In order to settle this problem, it is important to generalize the role of the net (ε), as used in Colombeau theory, into a more general ρ = (ρε) → 0 (which is called a gauge), and hence to generalize Re into some ρRe (see Def.1). We then introduce the set of hypernatural numbers as
ρ n ρ o Ne := [nε] ∈ Re | nε ∈ N ∀ε ,
1 so that it is natural to expect that n → 0 in the sharp topology if n → +∞ with n ∈ ρNe, because now n can also take infinite values. The notion of sequence σ ρ is therefore substituted with that of hypersequence, as a map (xn)n : Ne −→ Re, where σ is, generally speaking, another gauge. As we will see, (cf. example 33) only 1 ρ σ in this way we are able to prove e.g. that log n → 0 in Re as n ∈ Ne but only for a suitable gauge σ (depending on ρ), whereas this limit does not exist if σ = ρ. Finally, the notions of supremum and infimum are naturally linked to the notion of limit of a monotonic (hyper)sequence. Being an ordered set, ρRe already has a definition of, let us say, supremum as least upper bound. However, as already pre- liminary studied and proved by [9], this definition does not fit well with topological ρ ρ + properties of Re because generalized numbers [xε] ∈ Re can actually jump as ε → 0 (see Sec. 4.1). It is well known that in R we have m = sup(S) if and only if m is an upper bound of S and
∀r ∈ R>0 ∃s ∈ S : m − r ≤ s. (1.1) This could be generalized into the notion of close supremum in ρRe, generalizing [9], that results into better topological properties, see Sec.4. The ideas presented in the present article, which is self-contained, can surely be useful to explore similar ideas in other non-Archimedean settings, such as [7,6, 24, 19, 15].
2. The Ring of Robinson Colombeau and the hypernatural numbers In this section, we introduce our non-Archimedean ring of scalars and its subset of hypernatural numbers. For more details and proofs about the basic notions introduced here, the reader can refer e.g. to [8, 13, 14]. As we mentioned above, in order to accomplish the theory of hyperlimits, it is important to generalize Colombeau generalized numbers by taking an arbitrary asymptotic scale instead of the usual ρε = ε: I + Definition 1. Let ρ = (ρε) ∈ (0, 1] be a net such that (ρε) → 0 as ε → 0 (in the following, such a net will be called a gauge), then −a (i) I(ρ) := {(ρε ) | a ∈ R>0} is called the asymptotic gauge generated by ρ. (ii) If P(ε) is a property of ε ∈ I, we use the notation ∀0ε : P(ε) to denote 0 ∃ε0 ∈ I ∀ε ∈ (0, ε0]: P(ε). We can read ∀ ε as for ε small. SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 3
I (iii) We say that a net (xε) ∈ R is ρ-moderate, and we write (xε) ∈ Rρ if + ∃(Jε) ∈ I(ρ): xε = O(Jε) as ε → 0 , i.e., if 0 −N ∃N ∈ N ∀ ε : |xε| ≤ ρε . I (iv) Let (xε), (yε) ∈ R , then we say that (xε) ∼ρ (yε) if −1 + ∀(Jε) ∈ I(ρ): xε = yε + O(Jε ) as ε → 0 , that is if 0 n ∀n ∈ N ∀ ε : |xε − yε| ≤ ρε . This is a congruence relation on the ring Rρ of moderate nets with respect to pointwise operations, and we can hence define ρ Re := Rρ/ ∼ρ, which we call Robinson-Colombeau ring of generalized numbers. This name is justified by [21,8]: Indeed, in [21] A. Robinson introduced the notion of moderate and negligible nets depending on an arbitrary fixed infinitesimal ρ (in the framework of nonstandard analysis); independently, J.F. Colombeau, cf. e.g. [8] and references therein, studied the same concepts without using nonstandard analysis, but considering only the particular infinitesimal (ε). (v) In particular, if the gauge ρ = (ρε) is non-decreasing, then we say that ρ is a monotonic gauge. Clearly, considering a monotonic gauge narrows the class of moderate nets: e.g. if lim 1 xε = +∞ for all k ∈ N>0, then (xε) ∈/ Rρ for ε→ k any monotonic gauge ρ. In the following, ρ will always denote a net as in Def.1, even if we will sometimes omit the dependence on the infinitesimal ρ, when this is clear from the context. We will also use other directed sets instead of I: e.g. J ⊆ I such that 0 is a closure point of J, or I × N. The reader can easily check that all our constructions can be repeated in these cases. I We also recall that we write [xε] ≤ [yε] if there exists (zε) ∈ R such that (zε) ∼ρ 0 (we then say that (zε) is ρ-negligible) and xε ≤ yε + zε for ε small. Equivalently, we have that x ≤ y if and only if there exist representatives [xε] = x and [yε] = y such that xε ≤ yε for all ε. Although the order ≤ is not total, we still have the possibility to define the infimum [xε] ∧ [yε] := [min(xε, yε)], the supremum [xε] ∨ [yε] := [max(xε, yε)] of a finite number of generalized numbers. Henceforth, we will also use the customary notation ρRe∗ for the set of invertible generalized numbers, and we write x < y to say that x ≤ y and x − y ∈ ρRe∗. Our notations for intervals are: [a, b] := {x ∈ ρRe | ρ a ≤ x ≤ b},[a, b]R := [a, b] ∩ R. Finally, we set dρ := [ρε] ∈ Re, which is a positive −1 −1 invertible infinitesimal, whose reciprocal is dρ = [ρε ], which is necessarily a strictly positive infinite number. The following result is useful to deal with positive and invertible generalized numbers. For its proof, see e.g. [3,2, 13, 14]. Lemma 2. Let x ∈ ρRe. Then the following are equivalent: (i) x is invertible and x ≥ 0, i.e. x > 0. 0 (ii) For each representative (xε) ∈ Rρ of x we have ∀ ε : xε > 0. 0 m (iii) For each representative (xε) ∈ Rρ of x we have ∃m ∈ N ∀ ε : xε > ρε . 0 m (iv) There exists a representative (xε) ∈ Rρ of x such that ∃m ∈ N ∀ ε : xε > ρε . 4 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO
2.1. The language of subpoints. The following simple language allows us to simplify some proofs using steps recalling the classical real field R. We first introduce the notion of subpoint:
Definition 3. For subsets J, K ⊆ I we write K ⊆0 J if 0 is an accumulation point of K and K ⊆ J (we read it as: K is co-final in J). Note that for any J ⊆0 I, the constructions introduced so far in Def.1 can be repeated using nets ρ n (xε)ε∈J . We indicate the resulting ring with the symbol Re |J . More generally, no peculiar property of I = (0, 1] will ever be used in the following, and hence all the presented results can be easily generalized considering any other directed set. If ρ n 0 ρ n 0 K ⊆0 J, x ∈ Re |J and x ∈ Re |K , then x is called a subpoint of x, denoted as 0 0 0 0 x ⊆ x, if there exist representatives (xε)ε∈J ,(xε)ε∈K of x, x such that xε = xε 0 0 for all ε ∈ K. In this case we write x = x|K , dom(x ) := K, and the restriction ρ n ρ n ρ n (−)|K : Re −→ Re |K is a well defined operation. In general, for X ⊆ Re we set ρ n X|J := {x|J ∈ Re |J | x ∈ X}. In the next definition, we introduce binary relations that hold only on subpoints. Clearly, this idea is inherited from nonstandard analysis, where cofinal subsets are always taken in a fixed ultrafilter.
ρ Definition 4. Let x, y ∈ Re, L ⊆0 I, then we say
(i) x x ≤ y ⇐⇒ ∀L ⊆0 I : x ≤L y x < y ⇐⇒ ∀L ⊆0 I : x Note explicitly that, generally speaking, relations on subpoints, such as ≤s or =s, do not inherit the same properties of the corresponding relations for points. So, e.g., both =s and ≤s are not transitive relations. The next result clarifies how to equivalently write a negation of an inequality or of an equality using the language of subpoints. Lemma 5. Let x, y ∈ ρRe, then (i) x y ⇐⇒ x >s y (ii) x 6< y ⇐⇒ x ≥s y (iii) x 6= y ⇐⇒ x >s y or x Proof. (i) ⇐: The relation x >s y means x|L > y|L for some L ⊆0 I. By Lem.2 ρ 0 for the ring Re|L, we get ∀ ε ∈ L : xε > yε, where x = [xε], y = [yε] are any representatives of x, y resp. The conclusion follows by (2.1) SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 5 (i) ⇒: Take any representatives x = [xε], y = [yε]. The property 0 q ∀q ∈ R>0 ∀ ε : xε ≤ yε + ρε for q → +∞ implies x ≤ y. We therefore have q ∃q ∈ R>0 ∃L ⊆0 I ∀ε ∈ L : xε > yε + ρε, i.e. x >L y. (ii) ⇒: We have two cases: either x − y is not invertible or x 6≤ y. In the former case, the conclusion follows from [14, Thm. 1.2.39]. In the latter one, it follows from (i). (ii) ⇐: By contradiction, if x < y then x =L y for some L ⊆0 I, which contradicts the invertibility of x − y. (iii) ⇒: By contradiction, assume that x 6>s y and x 6 Lemma 6. Let x, y ∈ ρRe, then (i) x ≤ y or x >s y (ii) ¬(x >s y and x ≤ y) (iii) x = y or x ρ As usual, we note that these results can also be trivially repeated for the ring Re|L. So, e.g., we have x 6≤L y if and only if ∃J ⊆0 L : x >J y, which is the analog of ρ Lem.5.(i) for the ring Re|L. The second form of trichotomy (which for ρRe can be more correctly named as quadrichotomy) is stated as follows: ρ Lemma 7. Let x = [xε], y = [yε] ∈ Re, then c (i) x ≤ y or x ≥ y or ∃L ⊆0 I : L ⊆0 I, x ≥L y and x ≤Lc y (ii) If for all L ⊆0 I the following implication holds 0 x ≤L y, or x ≥L y ⇒ ∀ ε ∈ L : P{xε, yε} , (2.2) 0 then ∀ ε : P{xε, yε}. (iii) If for all L ⊆0 I the following implication holds 0 x Proof. (i): if x 6≤ y, then x >s y from Lem.5.(i). Let [ xε] = x and [yε] = y be two representatives, and set L := {ε ∈ I | xε ≥ yε}. The relation x >s y implies that L ⊆0 I. Clearly, x ≥L y (but note that in general we cannot prove x >L y). If c c L 6⊆0 I, then (0, εo] ⊆ L for some ε0, i.e. x ≥ y. On the contrary, if L ⊆0 I, then x ≤Lc y. (ii): Property (i) states that we have three cases. If xε ≤ yε for all ε ≤ ε0, then it suffices to set L := (0, ε0] in (2.2) to get the claim. Similarly, we can proceed if x ≥ y. Finally, if x ≥L y and x ≤Lc y, then we can apply (2.2) both with L and Lc to obtain 0 ∀ ε ∈ L : P{xε, yε} 0 c ∀ ε ∈ L : P{xε, yε} , from which the claim directly follows. (iii): By contradiction, assume ∀ε ∈ L : ¬P {xε, yε} , (2.4) ρ for some L ⊆0 I. We apply (i) to the ring Re|L to obtain the following three cases: c x ≤L y or x ≥L y or ∃J ⊆0 L : J ⊆0 L, x ≥J y and x ≤J c y. (2.5) ρ If x ≤L y, by Lem.6.(iv) for the ring Re|L, this case splits into two sub-cases: x =L y or ∃K ⊆0 L : x ρ Definition 8. Let x = [xε] ∈ Re be a generalized number, then: (i) If ∃L ∈ R : L ≤ x, then xi := infe∈(0,ε] xe is called the inferior part of x. h i (ii) If ∃U ∈ R : x ≤ U, then xs := supe∈(0,ε] xe is called the superior part of x. Moreover, we set: ◦ (iii) xi := lim infε→0+ xε ∈ R ∪ {±∞}, where [xε] = x is any representative of x, is called the inferior standard part of x. Note that if ∃xi, i.e. if x is finitely ◦ ◦ ◦ bounded from below, then (xi) = xi ∈ R and xi ≥ xi. ◦ (iv) xs := lim supε→0+ xε ∈ R ∪ {±∞}, where [xε] = x is any representative of x, is called the superior standard part of x. Note that if ∃xs, i.e. if x is finitely ◦ ◦ ◦ bounded from above, then (xs) = xs ∈ R and xs ≤ xs. 0 Note that, since ρ = (ρε) is non-decreasing, if [xε] = x is another representative, 0 n n n then for all e ∈ (0, ε], we have xe ≤ xe + ρe ≤ xe + ρε ≤ ρε + supe∈(0,ε] xe 0 n and hence supe∈(0,ε] xe ≤ ρε + supe∈(0,ε] xe. This shows that inferior and supe- rior parts, when they exist, are well-defined. Moreover, if (zε) is negligible, then SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 7 ◦ lim supε→0+ (xε + zε) ≤ lim supε→0+ xε + 0, which shows that xs is well-defined ◦ (similarly for xi using super-additivitiy of lim inf). ◦ ◦ Clearly, xi ≤ x ≤ xs and xi ≤ xs . We have that the generalized number x is ◦ ◦ ◦ near-standard if and only if xi = xs =: x ∈ R; it is infinitesimal if and only if ◦ ◦ ◦ ◦ ∃ x = 0; it is a positive infinite number if and only if xi = xs =: x = +∞ (the ◦ ◦ same for negative infinite numbers); it is a finite number if and only if xi , xs ∈ R. 0 00 0 ◦ 00 ◦ Finally, there always exist x , x ⊆ x such that x ≈ xi and x ≈ xs , where x ≈ y means that x − y is infinitesimal (i.e. |x − y| ≤ r for all r ∈ R>0 or, equivalently, limε→0+ xε − yε = 0 for all [xε] = x,[yε] = y). Therefore, any generalized number in ρRe is either finite or some of its subpoints is infinite; in the former case, some of its subpoints is near standard. 2.3. Topologies on ρRen. On the ρRe-module ρRen we can consider the natural ex- ρ ρ n tension of the Euclidean norm, i.e. |[xε]| := [|xε|] ∈ Re, where [xε] ∈ Re . Even if this generalized norm takes values in ρRe, it shares some essential properties with classical norms: |x| = x ∨ (−x) |x| ≥ 0 |x| = 0 ⇒ x = 0 |y · x| = |y| · |x| |x + y| ≤ |x| + |y| ||x| − |y|| ≤ |x − y|. It is therefore natural to consider on ρRen topologies generated by balls defined by this generalized norm and a set of radii: Definition 9. We say that R is a set of radii if ρ ∗ (i) R ⊆ Re≥0 is a non-empty subset of positive invertible generalized numbers. (ii) For all r, s ∈ R the infimum r ∧ s ∈ R. (iii) k · r ∈ R for all r ∈ R and all k ∈ R>0. Moreover, if R is a set of radii and x, y ∈ ρRe, then: (i) We write x ρ ∗ For example, Re≥0 and R>0 are sets of radii. Lemma 10. Let R be a set of radii and x, y, z ∈ ρRe, then (i) ¬ (x The relation n R ρ no Theorem 11. The set of balls Br (x) | r ∈ R, x ∈ Re generated by a set of radii R is a base for a topology on ρRen. ρ ∗ ρ Henceforth, we will only consider the sets of radii Re≥0 = Re>0 and R>0 and will use R ρ the simplified notation Br(x) := Br (x) if R = Re>0. The topology generated in the former case is called sharp topology, whereas the latter is called Fermat topology. We will call sharply open set any open set in the sharp topology, and large open set any open set in the Fermat topology; clearly, the latter is coarser than the former. It is well-known (see e.g. [2,3, 10, 11, 13] and references therein) that this is an equivalent way to define the sharp topology usually considered in the ring of Colombeau generalized numbers. We therefore recall that the sharp topology on ρRen is Hausdorff and Cauchy complete, see e.g. [2, 11]. 2.4. Open, closed and bounded sets generated by nets. A natural way to ρ n obtain sharply open, closed and bounded sets in Re is by using a net (Aε) of n subsets Aε ⊆ R . We have two ways of extending the membership relation xε ∈ Aε ρ n to generalized points [xε] ∈ Re (cf. [18, 12]): n Definition 12. Let (Aε) be a net of subsets of R , then n ρ n 0 o (i)[ Aε] := [xε] ∈ Re | ∀ ε : xε ∈ Aε is called the internal set generated by the net (Aε). n (ii) Let (xε) be a net of points of R , then we say that xε ∈ε Aε, and we read it as (xε) strongly belongs to (Aε), if 0 (i) ∀ ε : xε ∈ Aε. 0 0 (ii) If(xε) ∼ρ (xε), then also xε ∈ Aε for ε small. n ρ n o Moreover, we set hAεi := [xε] ∈ Re | xε ∈ε Aε , and we call it the strongly internal set generated by the net (Aε). (iii) We say that the internal set K = [Aε] is sharply bounded if there exists ρ M ∈ Re>0 such that K ⊆ BM (0). (iv) Finally, we say that the net (Aε) is sharply bounded if there exists N ∈ R>0 0 −N such that ∀ ε ∀x ∈ Aε : |x| ≤ ρε . Therefore, x ∈ [Aε] if there exists a representative [xε] = x such that xε ∈ Aε for ε small, whereas this membership is independent from the chosen representative in case of strongly internal sets. An internal set generated by a constant net Aε = A ⊆ Rn will simply be denoted by [A]. The following theorem (cf. [18, 12] for the case ρε = ε, and [13] for an arbi- trary gauge) shows that internal and strongly internal sets have dual topological properties: n n Theorem 13. For ε ∈ I, let Aε ⊆ R and let xε ∈ R . Then we have 0 q (i) [xε] ∈ [Aε] if and only if ∀q ∈ R>0 ∀ ε : d(xε,Aε) ≤ ρε. Therefore [xε] ∈ [Aε] ρ if and only if [d(xε,Aε)] = 0 ∈ Re. 0 c q c n (ii) [xε] ∈ hAεi if and only if ∃q ∈ R>0 ∀ ε : d(xε,Aε) > ρε, where Aε := R \Aε. c c Therefore, if (d(xε,Aε)) ∈ Rρ, then [xε] ∈ hAεi if and only if [d(xε,Aε)] > 0. (iii) [Aε] is sharply closed. (iv) hAεi is sharply open. n (v) [Aε] = [cl (Aε)], where cl (S) is the closure of S ⊆ R . n (vi) hAεi = hint(Aε)i, where int (S) is the interior of S ⊆ R . SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 9 For example, it is not hard to show that the closure in the sharp topology of a ball of center c = [cε] and radius r = [rε] > 0 is n o h i B (c) = x ∈ ρRd | |x − c| ≤ r = BE (c ) , (2.6) r e rε ε whereas n o B (c) = x ∈ ρRd | |x − c| < r = hBE (c )i. r e rε ε Using internal sets and adopting ideas similar to those used in proving Lem.7, we also have the following form of dichotomy law: n ρ n Lemma 14. For ε ∈ I, let Aε ⊆ R and let x = [xε] ∈ Re . Then we have: c c c (i) x ∈ [Aε] or x ∈ [Aε] or ∃L ⊆0 I : L ⊆0 I, x ∈L [Aε], x ∈Lc [Aε] (ii) If for all L ⊆0 I the following implication holds c 0 x ∈L [Aε] or x ∈L [Aε] ⇒ ∀ ε ∈ L : P{xε}, 0 then ∀ ε : P{xε}. c Proof. (i): If x∈ / [Aε], then xε ∈ Aε for all ε ∈ K and for some K ⊆0 I. Set c L := {ε ∈ I | xε ∈ Aε}, so that K ⊆ L ⊆0 I. We have x ∈L [Aε]. If L 6⊆0 I, then c c (0, ε0] ⊆ L for some ε0, i.e. x ∈ [Aε]. On the contrary, if L ⊆0 I, then x ∈Lc [Aε]. (ii): We can proceed as in the proof of Lem.7.(ii) using (i). 3. Hypernatural numbers We start by defining the set of hypernatural numbers in ρRe and the set of ρ- moderate nets of natural numbers: Definition 15. We set ρ n ρ o (i) Ne := [nε] ∈ Re | nε ∈ N ∀ε (ii) Nρ := {(nε) ∈ Rρ | nε ∈ N ∀ε} . ρ Therefore, n ∈ Ne if and only if there exists (xε) ∈ Rρ such that n = [int(|xε|)]. Clearly, N ⊂ ρNe. Note that the integer part function int(−) is not well-defined on ρ h 1/εi h 1/εi 1/ε Re. In fact, if x = 1 = 1 − ρε = 1 + ρε , then int 1 − ρε = 0 whereas 1/ε int 1 + ρε = 1, for ε sufficiently small. Similar counter examples can be set for floor and ceiling functions. However, the nearest integer function is well defined on ρNe, as proved in the following Lemma 16. Let (nε) ∈ Nρ and (xε) ∈ Rρ be such that [nε] = [xε]. Let rpi : R −→ N be the function rounding to the nearest integer with tie breaking towards positive 1 infinity, i.e. rpi(x) = bx + 2 c. Then rpi(xε) = nε for ε small. The same result holds using rni : R −→ N, the function rounding half towards −∞. 1 Proof. We have rpi(x) = bx + 2 c, where b−c is the floor function. For ε small, 1 1 ρε < 2 and, since [nε] = [xε], always for ε small, we also have nε − ρε + 2 < 1 1 1 1 xε + 2 < nε + ρε + 2 . But nε ≤ nε − ρε + 2 and nε + ρε + 2 < nε + 1. Therefore 1 bxε + 2 c = nε. An analogous argument can be applied to rni(−). Actually, this lemma does not allow us to define a nearest integer function ni : ρ Ne −→ Nρ as ni([xε]) := rpi(xε) because if [xε] = [nε], the equality nε = rpi(xε) holds only for ε small. A simpler approach is to choose a representative (nε) ∈ Nρ 10 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO ρ for each x ∈ Ne and to define ni(x) := (nε). Clearly, we must consider the net (ni(x)ε) only for ε small, such as in equalities of the form x = [ni(x)ε]. This is what we do in the following Definition 17. The nearest integer function ni(−) is defined by: ρ (i) ni : Ne :−→ Nρ ρ 0 (ii) If [xε] ∈ Ne and ni ([xε]) = (nε) then ∀ ε : nε = rpi(xε). ρ In other words, if x ∈ Ne, then x = [ni(x)ε] and ni(x)ε ∈ N for all ε. Another possibility is to formulate Lem. 16 as ρ [xε] ∈ Ne ⇐⇒ [xε] = [rpi(xε)]. Therefore, without loss of generality we may always suppose that xε ∈ N whenever ρ [xε] ∈ Ne. Remark 18. (i) σNe, with the order ≤ induced by σRe, is a directed set; it is closed with respect to sum and product although recursive definitions using σNe are not possible. (ii) In σNe we can find several chains (totally ordered subsets) such as: N, N · −k −k [int(ρε )] for a fixed k ∈ N, {[int(ρε )] | k ∈ N}. ρ n ρ nε (iii) Generally speaking, if m, n ∈ Ne, m ∈/ Ne because the net (mε ) can grow −K faster than any power (ρε ). However, if we take two gauges σ, ρ satisfying −1 σ ≤ ρ, using the net σε we can measure infinite nets that grow faster than −K −1 −1 σ (ρε ) because σε ≥ ρε for ε small. Therefore, we can take m, n ∈ Ne such that (ni(m)ε), (ni(n)ε) ∈ Rρ; we think at m, n as σ-hypernatural numbers growing at most polynomially with respect to ρ. Then, it is not hard to prove that if ρ is an arbitrary gauge, and we consider the auxiliary gauge e1/ρε n σ σε := ρε . then m ∈ Ne. ρ m mε (iv) If m ∈ Ne, then 1 := [(1 + zε) ], where (zε) is ρ-negligible, is well defined m mε and 1 = 1. In fact, log(1 + zε) is asymptotically equal to mεzε → 0, mε mε and this shows that ((1 + zε) ) is moderate. Finally, |(1 + zε) − 1| ≤ mε−1 |zε| mε(1 + zε) by the mean value theorem. 4. Supremum and Infimum in ρRe To solve the problems we explained in the introduction of this article, it is important to generalize at least two main existence theorems for limits: the Cauchy criterion and the existence of a limit of a bounded monotone sequence. The latter is clearly related to the existence of supremum and infimum, which cannot be always guaranteed in the non-Archimedean ring ρRe. As we will see more clearly later (see also [9]), to arrive at these existence theorems, the notion of supremum, i.e. the least upper bound, is not the correct one. More appropriately, we can associate a notion of close supremum (and close infimum) to every topology generated by a set of radii (see Def.9). Definition 19. Let R be a set of radii and let τ be the topology on ρRe generated by R. Let P ⊆ ρRe, then we say that τ separates points of P if ∀p, q ∈ P : p6=q ⇒ ∃A, B ∈ τ : p ∈ A, q ∈ B,A ∩ B = ∅, i.e. if P with the topology induced by τ is Hausdorff. SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 11 Definition 20. Let τ be a topology on ρRe generated by a set of radii R that separates points of P ⊆ ρRe and let S ⊆ ρRe. Then, we say that σ is (τ, P )-supremum of S if (i) σ ∈ P ; (ii) ∀s ∈ S : s ≤ σ; (iii) σ is a point of closure of S in the topology τ, i.e. if ∀A ∈ τ : σ ∈ A ⇒ ∃s¯ ∈ S ∩ A. Similarly, we say that ι is (τ, P )-infimum of S if (i) ι ∈ P ; (ii) ∀s ∈ S : ι ≤ s; (iii) ι is a point of closure of S in the topology τ, i.e. if ∀A ∈ τ : ι ∈ A ⇒ ∃s¯ ∈ S ∩ A. In particular, if τ is the sharp topology and P = ρRe, then following [9], we simply call the (τ, P )-supremum, the close supremum (the adjective close will be omitted if it will be clear from the context) or the sharp supremum if we want to underline the dependency on the topology. Analogously, if τ is the Fermat topology and P = R, then we call the (τ, P )-supremum the Fermat supremum. Note that (iii) implies that if σ is (τ, P )-supremum of S, then necessarily S 6= ∅. Remark 21. (i) Let S ⊆ ρRe, then from Def.9 and Thm. 11 we can prove that σ is the (τ, P )- supremum of S if and only if (a) ∀s ∈ S : s ≤ σ; (b) ∀r ∈ R ∃s¯ ∈ S : σ − r ≤ s¯. In particular, for the sharp supremum, (b) is equivalent to ∀q ∈ N ∃s¯ ∈ S : σ − dρq ≤ s.¯ (4.1) In the following of this article, we will also mainly consider the sharp topology and the corresponding notions of sharp supremum and infimum. (ii) If there exists the sharp supremum σ of S ⊆ ρRe and σ∈ / S, then from (4.1) it follows that S is necessarily an infinite set. In fact, applying (4.1) q1 with q1 := 1 we get the existence ofs ¯1 ∈ S such that σ − dρ < s¯1. We haves ¯1 6= σ because σ∈ / S. Hence, Lem.5.(iii) and Def. 20.(ii) yield that q2 s¯1 Proof. Assume that σ1 and σ2 are supremum of S. That is Def. 20.(ii) and (4.1) hold q both for σ1, σ2. Then, for all fixed q ∈ N, there existss ¯2 ∈ S such that σ2−dρ ≤ s¯2. q Hences ¯2 ≤ σ1 becauses ¯2 ∈ S. Analogously, we have that σ1 − dρ ≤ s¯1 ≤ σ2 for q q somes ¯1 ∈ S. Therefore, σ2 − dρ ≤ σ1 ≤ σ2 + dρ , and this implies σ1 = σ2 since q ∈ N is arbitrary. In [9], the notation sup(S) is used for the close supremum. On the other hand, we will never use the notion of supremum as least upper bound. For these reasons, 12 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO we prefer to use the simpler notation sup(S). Similarly, we use the notation inf(S) for the close (or sharp) infimum. From Rem. 21.(a) and (b) it follows that inf(S) = − sup(−S) (4.2) in the sense that the former exists if and only if the latter exists and in that case they are equal. For this reason, in the following we only study the supremum. Example 23. ρ (i) Let K = [Kε] bf Re be a functionally compact set (cf. [11]), i.e. K ⊆ BM (0) ρ for some M ∈ Re>0 and Kε b R for all ε. We can then define σε := sup(Kε) ∈ Kε. From K ⊆ BM (0), we get σ := [σε] ∈ K. It is not hard to prove that σ = sup(K) = max(K). Analogously, we can prove the existence of the sharp minimum of K. (ii) If S = (a, b), where a, b ∈ ρRe and a ≤ b, then sup(S) = b and inf(S) = a. n 1 ρ o (iii) If S = n | n ∈ Ne , then inf(S) = 0. (iv) Like in several other non-Archimedean rings, both sharp supremum and infi- mum of the set D∞ of all infinitesimals do not exist. In fact, by contradiction, if σ were the sharp supremum of D∞, then from (4.1) for q = 1 we would get ¯ ¯ the existence of h ∈ D∞ such that σ ≤ h + dρ. But then σ ∈ D∞, so also 2σ ∈ D∞. Therefore, we get 2σ ≤ σ because σ is an upper bound of D∞, and hence σ = 0 ≥ dρ, a contradiction. Similarly, one can prove that there does not exist the infimum of this set. (v) Let S = (0, 1)R = {x ∈ R | 0 < x < 1}, then clearly σ = 1 is the Fermat supre- mum of S whereas there does not exist the sharp supremum of S. Indeed, if σ = sup(S), then s ≤ σ ≤ s¯ + dρ for all s ∈ S and for somes ¯ ∈ S. Taking any s ∈ (¯s, 1)R ⊆ S we get s ≤ σ ≤ s¯ + dρ, which, for ε → 0, implies s ≤ s¯ because s,s ¯ ∈ R. This contradicts s ∈ (¯s, 1). In particular, 1 is not the sharp supremum. This example shows the importance of Def. 20, i.e. that the best notion of supremum in a non-Archimedean setting depends on a fixed topology. 1 c (vi) Let S = (0, 1) ∪ {sˆ} wheres ˆ|L = 2,s ˆ|Lc = 2 , L ⊆0 I, L ⊆0 I, then @ sup(S). In fact, if ∃σ := sup(S), then σ|L ≥ sˆ|L = 2 and σ|Lc = 1. Assume that ∃s¯ ∈ S : σ − dρ ≤ s¯, then 2 − dρ|L ≤ σ|L − dρ|L ≤ s¯|L. Thereby, 3 s¯|L > 2 and hences ¯ 6∈ (0, 1) ands ¯ =s ˆ. We hence get σ|Lc − dρ|Lc ≤ sˆ|Lc , 1 i.e. 1 − dρ|Lc ≤ 2 , which is impossible. We can intuitively say that the subpoints ˆ|L creates a “ε-hole” (i.e. a “hole” only for some ε) on the right of S and hence S is not “an ε-continuum” on this side. Finally note that the point u|L := 2 and u|Lc := 1 is the least upper bound of S. Lemma 24. Let A, B ⊆ ρRe, then ρ (i) ∀λ ∈ Re>0 : sup(λA) = λ sup(A), in the sense that one supremum exists if and only if the other one exists, and in that case they coincide; ρ (ii) ∀λ ∈ Re<0 : sup(λA) = λ inf(A), in the sense that one supremum/infimum exists if and only if the other one exists, and in that case they coincide; Moreover, if ∃ sup(A), sup(B), then: (iii) If A ⊆ B, then sup(A) ≤ sup(B); (iv) sup(A + B) = sup(A) + sup(B); ρ (v) If A, B ⊆ Re≥0, then sup(A · B) = sup(A) · sup(B). SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 13 1 Proof. (i): If ∃ sup(λA), then we have a ≤ λ sup(λA) for all a ∈ A. For all q ∈ N, we q 1 1 q can finda ¯ ∈ A such that sup(λA) − λa¯ ≤ dρ . Thereby, λ sup(λA) − a¯ ≤ λ dρ → 0 1 as q → +∞ because λ is moderate. This proves that ∃ sup(A) = λ sup(λA). Similarly, we can prove the opposite implication. (ii): From (i) and (4.2) we get: sup(λA) = sup(−λ(−A)) = −λ sup(−A) = λ inf(A). (iii): By contradiction, using Lem.5.(i), if sup( A) >L sup(B) for some L ⊆0 I, q ρ then sup(A) − sup(B) >L dρ for some q ∈ N by Lem.2 for the ring Re|L. Property (4.1) yields sup(A) − dρq ≤ a¯ for somea ¯ ∈ A, anda ¯ ≤ sup(B) because A ⊆ B. q q q Thereby, sup(A) − sup(B) ≤ dρ , which implies dρ In the next section, we introduce in the non-Archimedean framework ρRe how to approximate sup(S) of S ⊆ ρRe using points of S and upper bounds, and the non-Archimedean analogous of the notion of upper bound. 4.1. Approximations of Sup and Archimedean upper bounds. In the real field, we have the following peculiar properties: (i) The notion of least upper bound coincides with that of close supremum, i.e. it satisfies property (1.1). We can hence question when these two notions coincide also in ρRe. Example 23.(vi) shows that the answer is not trivial. A first solution of this problem is already contained in [9, Prop. 1.4], where it is shown that the close supremum, assuming that it exists, coincides with the least upper bound. (ii) The notion of upper bound in R is very useful because it entails the existence of the supremum. Clearly, since there are infinite upper bounds but only one supremum, the notion of upper bound results to be really useful in estimates with inequalities. Moreover, in the ring ρRe, the presence of infinite numbers (of different magnitudes) allows one to have trivial upper bounds, such as in the case S = (0, 1) and M = dρ−1, or S = (0, dρ−1) and M = dρ−2. Therefore, we can also investigate whether we can consider non trivial upper bounds, i.e. numbers which are, intuitively, of the same order of magnitude of the elements of S ⊆ ρRe. On the other hand, example 23.(vi) shows that with respect to any reasonable definition of “same order of magnitude”, the upper bound m = 3 must be of the same order of any point in S, although @ sup(S). We will solve this problem by introducing the definition of Archimedean upper bound. (iii) If ∅ 6= S ⊆ R admits an upper bound, then sup(S) can be arbitrarily ap- proximated using upper bounds and points of S. When is this possible if ∅= 6 S ⊆ ρRe? Example 23.(vi) shows that these problems cannot be solved in general, and we are hence searching for a useful sufficient condition on S. We first prove the following useful characterization of the existence of sup(S), which also solves problem (iii): Theorem 25. Let S ⊆ ρRe, and let U ⊆ ρRe denote the set of upper bounds of S. Then S has supremum if and only if q ∀q ∈ N ∃uq ∈ U ∃sq ∈ S : uq − sq ≤ dρ . (4.3) 14 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO Proof. If σ = sup(S), then (4.3) simply follows by setting uq := σ and sq ∈ S from (4.1). Vice versa, if (4.3) holds, then q q+1 −dρ ≤ sq − uq ≤ uq+1 − uq ≤ uq+1 − sq+1 ≤ dρ ∀q ∈ N. q p min(p,q)−1 Thereby, −(p−q)dρ ≤ up−uq ≤ (p−q)dρ for all p > q, and hence −dρ ≤ min(p,q)−1 up − uq ≤ dρ for all p, q ∈ N>0. This shows that (uq)q∈N is a Cauchy sequence which thus converges to some σ ∈ ρRe. Property (4.3) yields that also (sq)q∈N → σ, and this implies condition (4.1). Since each uq is an upper bound, for all s ∈ S we have s ≤ uq, which gives s ≤ σ for q → +∞. To solve problem (ii), assume that u ∈ ρRe is an upper bound of a non empty S ⊆ ρRe. Let [uε] = u, and for all s ∈ S choose a representative [sε] = s such that ∀ε ∈ I : sε ≤ uε. We can hence set σε := sup {sε | s ∈ S} ∀ε ∈ I. (4.4) Thereby, for all e ∈ I and for all q ∈ N q ∃s = s(e, q) ∈ S : σe − ρe < se = s(q, e)e. (4.5) If we think at example 23.(vi), for a fixed q ∈ N, the net (s(q, ε)ε) is not a repre- sentative of any point s ∈ S. This motivates the following definition (where we do not necessarily assume that S has an upper bound, as above). Definition 26. Let S ⊆ ρRe, then we say that S is complete from above if the following condition holds: assume that (s(q, e))q∈N is a family of S which is bounded e∈I from above: ρ ∃[uε] ∈ Re ∀e ∈ I ∀q ∈ N : s(q, e) ≤ [uε]; assume that we have chosen representatives [s(q, e)ε] = s(q, e) such that 0 ∀ ε ∀e ∈ I ∀q ∈ N : s(q, e)ε ≤ uε. (4.6) Then, we have 0 ∀q ∈ N ∃[¯sε] ∈ S ∀ ε : s(q, ε)ε ≤ s¯ε, (4.7) where S is the closure of S in the sharp topology. Moreover, if ∃s ∈ S : s > 0, then we say that M is an Archimedean upper bound (AUB) of S if (i) M ∈ ρRe and ∀s ∈ S : s ≤ M; (ii) ∃n ∈ N ∃s¯ ∈ S : M < ns¯. The minimum n ∈ N that satisfies this property is called the order of M (clearly, n ≥ 2). Note that this condition, using an Archimedean-like property, formalizes the idea that M ands ¯ are of the same order of magnitude. Dually, we can define the notion of completeness from below using a lower bound instead of [uε], and the inequality sε ≤ s(q, ε)ε. If ∃s ∈ S : s < 0, then N is an Archimedean lower bound (ALB) of S if it is a lower bound such that ∃n ∈ N ∃s¯ ∈ S :sn ¯ < N. Note that σ = sup(S) is always an AUB of order 2. In fact, from the q existence of s ∈ S>0, we have s > dρ for some q ∈ N and the existence ofs ¯ ∈ S withs ¯ ≥ σ − dρq+1. Thereby,s ¯ ≥ s − dρq+1 > dρq − dρq+1 > dρq+1 and thus σ ≤ s¯+ dρq+1 < 2¯s. We also note that S = ρRe is trivially complete from above (set ρ sε = uε) but6 ∃ sup( Re). The following results solve the remaining problems (i) and (ii). SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 15 Theorem 27. Assume that ∅= 6 S ⊆ ρRe is complete from above and that there exists an upper bound of S. Then (i) ∃ sup(S) =: σ; Let (sq)q∈N and (uq)q∈N be two sequences as in Thm. 25, then q (ii) If ∃s ∈ S : s > 0, and if there exists C ∈ R>0 such that sq ≥ Cdρ for all q ∈ N large, then uq is an AUB of S for all q sufficiently large; (iii) If ∃s ∈ S : s > 0, then uq is an AUB of S of order 2 for all q sufficiently large. Proof. For any s ∈ S, it cannot be σ Corollary 28. Let ∅= 6 S ⊆ ρRe. Assume that S is complete from above, then ∃ sup(S) if and only if S admits an upper bound. Now, we can also complete the relationships between close supremum and least upper bound. Corollary 29. Let ∅= 6 S ⊆ ρRe be complete from above, then ∃ sup(S) = σ ⇐⇒ σ is the least upper bound of S. q Proof. Let u be an upper bound of S; by condition (4.1) we get σ − dρ ≤ sq ≤ u for all q ∈ N and for some sq ∈ S. For q → +∞, we get σ ≤ u. Vice versa, if S 6= ∅ is complete from above and σ is the least upper bound of S, then the conclusion follows from Cor. 28. Example 30. (i) Example 23.(vi) shows that the assumption of being complete from above is necessary in Cor. 29. (ii) Any set having a maximum is trivially complete from above: set [¯sε] := max(S) in (4.7). 16 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO 1 ε (iii) S = (0, 1) is complete from above for [¯sε] = 1. Note that, considering ρε q instead of ρε in (4.5), we get [s(q, ε)ε] = 1 ∈ S \ S. (iv) There do not exist neither the supremum nor the least upper bound of S = 1 + D∞. On the other hand, 2 is an AUB of S and hence S is not complete from above. ρ r (v) D∞ has neither AUB nor ALB; Re has neither AUB nor ALB; {dρ | r ∈ R>0} has no supremum and no AUB and hence it is not complete from above. (vi) Assume that there does not exist and upper bound of S. This means that ρ ∀u ∈ Re ∃s ∈ S : s >s u. −q Thereby, there exists a sequence (sq)q∈N of S such that sq >s dρ . Based on this, we could set sup(S) := +∞. 5. The hyperlimit of a hypersequence 5.1. Definition and examples. Definition 31. A map x : σNe −→ ρRe, whose domain is the set of hypernatural numbers σNe is called a (σ−) hypersequence (of elements of ρRe). The values x(n) ∈ ρRe at n ∈σNe of the function x are called terms of the hypersequence and, as usual, denoted using an index as argument: xn = x(n). The hypersequence itself is denoted by (x ) , or simply (x ) if the gauge on the domain is clear from the n n∈σ Ne n n context. Let σ, ρ be two gauges, x : σNe −→ ρRe be a hypersequence and l ∈ ρRe. We σ say that l is hyperlimit of (xn)n as n→ ∞ and n∈ Ne, if σ σ q ∀q ∈ N ∃M ∈ Ne ∀n ∈ Ne≥M : |xn − l| < dρ . In the following, if not differently stated, ρ and σ will always denote two gauges ρ and (xn)n a σ-hypersequence of elements of Re. Finally, if σε ≥ ρε, at least for all ε small, we simply write σ ≥ ρ. ρ Remark 32. In the assumption of Def. 31, let k ∈ Re>0, N ∈ N, then the following are equivalent: ρ σ (i) l∈ Re is the hyperlimit of (xn)n as n ∈ Ne. ρ σ σ (ii) ∀η ∈ Re>0 ∃M ∈ Ne ∀n ∈ Ne≥M : |xn − l| < η. ρ σ σ (iii) Let U ⊆ Re be a sharply open set, if l ∈ U then ∃M ∈ Ne ∀n ∈ Ne≥M : xn ∈ U. σ σ q (iv) ∀q ∈ N ∃M ∈ Ne ∀n ∈ Ne≥M : |xn − l| < k · dρ . σ σ q−N (v) ∀q ∈ N ∃M ∈ Ne ∀n ∈ Ne≥M : |xn − l| < dρ . q+1 q Directly by the inequality |l1 − l2| ≤ |l1 − xn| + |l2 − xn| ≤ 2dρ < dρ (or by using that the sharp topology on ρRe is Hausdorff) it follows that there exists at most one hyperlimit, so that we can use the notation ρ lim xn := l. n∈σ Ne As usual, a hypersequence (not) having a hyperlimit is said to be (non-)convergent. σ ρ We can also similarly say that (xn)n : Ne −→ Re is divergent to +∞ (−∞) if σ σ −q −q ∀q ∈ N ∃M ∈ Ne ∀n ∈ Ne≥M : xn > dρ (x < −dρ ). SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 17 Example 33. (i) If σ ≤ ρR for some R ∈ R , we have ρ lim 1 = 0. In fact, 1 < dρq holds >0 n∈σ Ne n n −q σ −q −q/R e.g. if n > [int (ρε ) + 1] ∈ Ne because ρε ≤ σε for ε small. − 1 ρε (ii) Let ρ be a gauge and set σε := exp − ρε , so that σ is also a gauge. We have ρ 1 ρ ρ 1 lim = 0 ∈ Re whereas @ lim n∈σ Ne log n n∈ρNe log n 1 q −q In fact, if n > 1, we have 0 < log n < dρ if and only if log n > dρ , i.e. dρ−q ρ ρ−q σ n > e (in Re). We can thus take M := int e ε + 1 ∈ Ne because 1 −q − ρ ρε −1 e ε < exp ρε = σε for ε small. Vice versa, by contradiction, if ∃ρ lim 1 =: l ∈ ρR, then by the definition of hyperlimit from ρN to ρR, n∈ρNe log n e e e we would get the existence of M ∈ ρNe such that 1 1 ∀n ∈ ρNe : n ≥ M ⇒ − dρ < l < + dρ (5.1) log n log n We have to explore two possibilities: if l is not invertible, then lεk = 0 for some sequence (εk) ↓ 0 and some representative [lε] = l. Therefore from 31, we get 1 < lεk + ρεk = ρεk log Mεk 1 − ρ ρ εk hence Mεk > e ∀k ∈ N, in contradiction with M ∈ Re. If l is invertible, then dρp < |l| for some p ∈ N. Setting q := min{p ∈ N | dρp < |l|} + 1, we q get that lε¯k < ρε¯k for some sequence (¯εk)k ↓ 0. Therefore 1 q < lε¯k + ρε¯k ≤ |lε¯k | + ρε¯k < ρε¯k + ρε¯k log Mε¯k 1 and hence Mε¯ > exp q for all k ∈ N, which is in contradiction with k ρ +ρε ε¯k k M ∈ ρRe because q ≥ 1. " − 1 # ρ ρ Analogously, we can prove that ρ lim 1 = 0 if σ = [σ ] = e−e n∈σ Ne log(log n) whereas ρ lim 1 (and similarly using log(log(...... k (log n) ...). @ n∈ρNe log(log n) −n 1 ρ ρ (iii) Set xn := dρ if n ∈ N, and xn := n if n ∈ Ne \ N, then {xn | n ∈ Ne} is unbounded in ρR even if ρ lim x = 0. Similarly, if x := dρn if n ∈ N e n∈ρNe n n ρ and xn := sin(n) otherwise, then limn→+∞ xn = 0 whereas @ limn∈ρN xn. In n∈N e general, we can hence only state that convergent hypersequence are eventually bounded: ρ ρ σ σ ∃ lim xn ⇒ ∃M ∈ Re ∃N ∈ Ne ∀n ∈ Ne≥N : |xn| ≤ M. n∈σ Ne (iv) If k < 1 and k > 1, then ρ lim kn = 0 and ρ lim kn = +∞, hence s s n∈ρNe s n∈ρNe s ρ lim kn. @ n∈ρNe 18 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO n 2 (v) Since for n ∈ N we have (1−dρ) = 1−ndρ+On(dρ ), it is not hard to prove n n that ((1 − dρ) )n∈N is not a Cauchy sequence. Therefore, @ limn∈N(1 − dρ) , whereas ρ lim (1 − dρ)n = 0. n∈ρNe ρ A sufficient condition to extend an ordinary sequence (an)n∈N : N −→ Re of ρ-generalized numbers to the whole σNe is σ ∀n ∈ Ne : ani(n)ε ∈ Rρ. (5.2) ρ σ In fact, in this way an := ani(n)ε ∈ Re for all n ∈ Ne, is well-defined because of Lem. 16; on the other hand, we have defined an extension of the old sequence (an)n∈N because if n ∈ N, then ni(n)ε = n for ε small and hence an = [an]. For 1 −1 example, the sequence of infinities an = n + dρ for all n ∈ N can be extended to σ −n σ ρ any Ne, whereas an = dσ can be extended as a : Ne −→ Re only for some gauges ρ, e.g. if the gauges satisfy 0 n N ∃N ∈ N ∀n ∈ N ∀ ε : σε ≥ ρε , (5.3) 1/ε (e.g. σε = ε and ρε = ε ). The following result allows us to obtain hyperlimits by proceeding ε-wise Theorem 34. Let (an,ε)n,ε : N × I −→ R. Assume that for all ε ∃ lim an,ε =: lε, (5.4) n→+∞ ρ and that l := [lε] ∈ Re. Then there exists a gauge σ (not necessarily a monotonic one) such that σ σ ρ (i) There exists M ∈ Ne and a hypersequence (an)n : Ne −→ Re such that a = [a ] ∈ ρR for all n ∈ σN ; n ni(n)ε,ε e e≥M (ii) l = ρ lim a . n∈σ Ne n Proof. From (5.4), we have q q ∀ε ∀q ∃Mεq ∈ N>0 ∀n ≥ Mεq : ρε − lε < an,ε < ρε + lε. (5.5) Without loss of generality, we can assume to have recursively chosen Mεq so that Mεq ≤ Mε,q+1 ∀ε ∀q. (5.6) 0 1 Set M¯ ε := M 1 > 0; since ∀q ∈ N ∀ ε : q ≤ d e,(5.6) implies ε,d ε e ε 0 ∀q ∈ N ∀ ε : M¯ ε ≥ Mεq. (5.7) ¯ ¯ −1 If the net (Mε) is ρ-moderate, set σ := ρ, otherwise set σε := min ρε, Mε ∈ (0, 1]. + Thereby, the net σε → 0 as ε → 0 (note that not necessarily σ is non-decreasing, −1 e.g. if lim 1 M¯ ε = +∞ for all k ∈ N>0 and M¯ ε ≥ ρ ), i.e. it is a gauge. Now set ε→ k ε ¯ ¯ σ ¯ −1 σ M := [Mε] ∈ Ne because our definition of σ yields Mε ≤ σε , Mq := [Mεq] ∈ Ne because of (5.7), and ( [a ] if n ≥ M in σN ni(n)ε,ε 1 e σ an := ∀n ∈ Ne. (5.8) 1 otherwise σ ρ We have to prove that this well-defines a hypersequence (an)n : Ne −→ Re. First of all, the sequence is well-defined with respect to the equality in σNe because of Lem. 16. Moreover, setting q = 1 in (5.5), we get ρε − lε < an,ε < ρε + lε for all ε σ and for all n ≥ Mε1. If n ≥ M1 in Ne, then ni(n)ε ≥ Mε1 for ε small, and hence SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 19 ρ − l < a < ρ + l . This shows that a ∈ ρR because we assumed that ε ε ni(n)ε,ε ε ε n e ρ l = [lε] ∈ Re. Finally, (5.5) and (5.6) yield that if n ≥ Mq then n ≥ M1 and hence q |an − l| < dρ . From the proof it also follows, more generally, that if (Mεq)ε,q satisfies (5.5) and if ∃(qε) → +∞ :(Mε,qε ) ∈ Rρ, 1 then we can repeat the proof with qε instead of d ε e and setting σ := ρ. 5.2. Operations with hyperlimits and inequalities. Thanks to Def.9 of sharp topology and our notation for x < y (and of the consequent Lem.2), some results about hyperlimits can be proved by trivially generalizing classical proofs. For ex- ample, if (xn)n∈σ N and (yn)n∈σ N are two convergent hypersequences then their sum e e xn (xn + yn) σ , product (xn · yn) σ and quotient (the last one being n∈ Ne n∈ Ne yn σ n∈ Ne σ defined only when yn is invertible for all n∈ Ne) are convergent hypersequences and the corresponding hyperlimits are sum, product and quotient of the corresponding hyperlimits. The following results generalize the classical relations between limits and in- equalities. Theorem 35. Let x, y, z : σNe −→ ρRe be hypersequences, then we have: (i) If ρ lim x < ρ lim y , then ∃M∈ σN such that x < y for all n ≥ M, n∈σ Ne n n∈σ Ne n e n n n ∈ σNe. (ii) If x ≤ y ≤ z for all n ∈ σN and ρ lim x = ρ lim z =: l, then n n n e n∈σ Ne n n∈σ Ne n ∃ ρ lim y = l, n∈σ Ne n Proof. (i) follows from Lem.2 and the Def. 31 of hyperlimit. For (ii), the proof is analogous to the classical one. In fact, since ρ lim x = ρ lim z =: l given n∈σ Ne n n∈σ Ne n 0 00 σ q q q ∈ N, there exist M , M ∈ Ne such that l − dρ < xn and zn < l + dρ for all 0 00 σ 0 00 q n > M , n > M , n ∈ Ne, then for n > M := M ∨ M , we have l − dρ < xn ≤ q yn ≤ zn < l + dρ . Theorem 36. Assume that C is a sharply closed subset of ρR, that ∃ ρ lim x =: e n∈σ Ne n σ σ l and that xn eventually lies in C, i.e. ∃N ∈ Ne ∀n ∈ Ne≥N : xn ∈ C. Then also l ∈ C. In particular, if (y ) is another hypersequence such that ∃ ρ lim y =: k, n n n∈σ Ne n σ σ then ∃N ∈ Ne ∀n ∈ Ne≥N : xn ≥ yn implies l ≥ k. Proof. A reformulation of the usual proof applies. In fact, let us suppose that ρ ρ ρ l ∈ Re \ C. Since Re \ C is sharply open, there is an η > 0, for which Bη(l)⊆ Re\C. σ Letn ¯ ∈ Ne≥N be such that |xn − l| < η when n > n¯. Then we have xn ∈ C and ρ xn ∈ Bη(l) ⊆ Re \ C, a contradiction. The following result applies to all generalized smooth functions (and hence to all Colombeau generalized functions, see e.g. [12, 13]; see also [1] for a more general class of functions) because of their continuity in the sharp topology. Theorem 37. Suppose that f : U −→ ρRe. Then f is sharply continuous function at x = c if and only if it is hyper-sequentially continuous, i.e. for any hyperse- quence (xn)n in U converging to c, the hypersequence (f (xn))n converges to f (c), i.e. f ρ lim x = ρ lim f(x ). n∈σ Ne n n∈σ Ne n 20 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO Proof. We only prove that the hyper-sequential continuity is a sufficient condition, because the other implication is a trivial generalization of the classical one. By contradiction, assume that for some Q ∈ N n Q ∀n ∈ N ∃xn ∈ U : |xn − c| < dρ , |f(xn) − f(c)| >s dρ . (5.9) ρ −N For n ∈ N set ωn := n and for n ∈ Ne \ N set ωn := min N ∈ N | n ≤ dρ and ρ ωn xn := xωn . Then for all n ∈ Ne, from (5.9) we get |xn − c| < dρ → 0 because ρ ωn → +∞ as n → +∞ in n ∈ Ne. Therefore, (xn)n is an hypersequence of U that converges to c, which yields f(xn) → f(c), in contradiction with (5.9). R Example 38. Let σ ≤ ρ for some R ∈ R>0. The following inequalities hold for all generalized numbers because they also hold for all real numbers: ln(x) ≤ x nn nn e ≤ n! ≤ en . (5.10) e e √ √ From the first one it follows 0 ≤ ln(n) = 2 ln n ≤ 2 n , so that ρ lim ln(n) := 0 n n n n∈σ Ne n ρ 1/n ρ 1/n from Thm. 35 and limn∈σ N n = 1 from Thm. 37 and hence limn∈σ N (n!) = e n e +∞ by (5.10). Similarly, we have ρ lim 1 + 1 = e because n log 1 + 1 = n∈σ Ne n n 1 1 1 − 2n + O n2 → 1 and because of Thm. 37. A little more involved proof concerns L’Hˆopitalrule for generalized smooth func- tions. For the sake of completeness, here we only recall the equivalent definition: Definition 39. Let X ⊆ ρRen and Y ⊆ ρRed. We say that f : X −→ Y is a generalized smooth function (GSF) if (i) f : X −→ Y is a set-theoretical function. ∞ n d (0,1] (ii) There exists a net (fε) ∈ C (R , R ) such that for all [xε] ∈ X: (a) f(x) = [fε(xε)] n α (b) ∀α ∈ N :(∂ fε(xε)) is ρ − moderate. For generalized smooth functions lots of results hold: closure with respect to com- position, embedding of Schwartz’s distributions, differential calculus, one-dimensio- nal integral calculus using primitives, classical theorems (intermediate value, mean value, Taylor, extreme value, inverse and implicit function), multidimensional inte- gration, Banach fixed point theorem, a Picard-Lindel¨oftheorem for both ODE and PDE, several results of calculus of variations, etc. In particular, we have the following (see also [10] for the particular case of Colombeau generalized functions): Theorem 40. Let U ⊆ ρRe be a sharply open set and let f : U −→ ρRe be a GSF ∞ defined by the net of smooth functions fε ∈ C (R, R). Then ρ (i) There exists an open neighbourhood T of U × {0} and a GSF Rf : T → Re, called the generalized incremental ratio of f, such that f(x + h) = f(x) + h · Rf (x, h) ∀(x, h) ∈ T. (5.11) 0 0 Moreover Rf (x, 0) = [fε(xε)] = f (x) is another GSF and we can hence recursively define f (k)(x). (ii) Any two generalized incremental ratios of f coincide on the intersection of their domains. SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 21 (iii) More generally, for all k ∈ N>0 there exists an open neighbourhood T of k ρ U × {0} and a GSF Rf : T → Re, called k-th order Taylor ratio of f, such that k−1 X f (j)(x) f(x + h) = hj + Rk(x, h) · hk ∀(x, h) ∈ T. (5.12) j! f j=0 Any two ratios of f of the same order coincide on the intersection of their domains. We can now prove the following generalization of one of L’Hˆopitalrule: ρ σ Theorem 41. Let U ⊆ Re be a sharply open set (xn)n, (yn)n : Ne −→ U be hypersequences converging to l ∈ U and m ∈ U respectively and such that xn − l ρ lim =: C ∈ ρRe. n∈σ Ne yn − m ρ σ Let k ∈ N>0 and f, g : U −→ Re be GSF such that for all n ∈ Ne and all j = 0, . . . , k − 1 (j) ρ ∗ g (yn) ∈ Re f (j)(l) = g(j)(m) = 0 (5.13) g(k)(m) ∈ ρRe∗ Then for all j = 0, . . . , k − 1 f (j)(x ) f (k)(x ) ∃ ρ lim n = Ck · ρ lim n . (j) (k) n∈σ Ne g (yn) n∈σ Ne g (yn) Proof. Using (5.12) and (5.13), we can write (j) Pk−1 f (l) j k k f(x ) (xn − l) + (xn − l) R (l, xn − l) n = j=0 j! f = k−1 g(j)(m) g(yn) P j k k j=0 j! (yn − m) + (yn − m) Rg (m, yn − m) k k xn − l Rf (l, xn − l) = · k . yn − m Rg (m, yn − m) k k Since Rf and Rg are GSF, they are sharply continuous. Therefore, the right hand k (k) k Rf (l,0) k f (l) side of the previous equality tends to C · k = C · (k) . At the same limit Rg (m,0) g (m) (k) k f (xn) (k) (k) converges the quotient C (k) because f and g are also GSF and hence g (yn) they are sharply continuous. The claim for j = 1, . . . , k − 1 follows by applying the (j) (j) conclusion for j = 0 with f and g instead of f and g. Note that for xn = yn, l = m, we have C = 1 and we get the usual L’Hˆopitalrule (formulated using hypersequences). Note that a similar theorem can also be proved without hypersequences and using the same Taylor expansion argument as in the previous proof. 22 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO 5.3. Cauchy criterion and monotonic hypersequences. In this section, we deal with classical criteria implying the existence of a hyperlimit. Definition 42. We say that (x ) is a Cauchy hypersequence if n n∈σ Ne σ σ q ∀q ∈ N ∃M ∈ Ne ∀n, m ∈ Ne≥M : |xn − xm| < dρ . Theorem 43. A hypersequence converges if and only if it is a Cauchy hyperse- quence Proof. To prove that the Cauchy criterion is a necessary condition it suffices to consider the inequalities: q+1 q+1 q |xn − xm| ≤ |xn − l| + |xm − l| ≤ dρ + dρ < dρ Vice versa, assume that σ σ q ∀q ∈ N ∃Mq ∈ Ne ∀n, m ∈ Ne≥Mq : |xn − xm| < dρ . (5.14) ρ The idea is to use Cauchy completeness of Re. In fact, set h1 := M1 and hq+1 := Mq+1 ∨ hq. We claim that (xhq )q∈N is a standard Cauchy sequence converging to the same limit of (x ) . From (5.14) it follows that (x ) is a standard n n∈σ Ne hq q∈N Cauchy sequence (in the sharp topology). Therefore, there existsx ¯ ∈ ρRe such that limq→+∞ xhq =x ¯. Now, fix q ∈ N and pick any m ≥ q + 1 such that q+1 |xhm − x¯| < dρ . (5.15) Then for all N ≥ Mq+1 we have: q+1 q |xN − x¯| ≤ |xN − xhm | + |xhm − x¯| < 2dρ < dρ because hm ≥ hq+1 ≥ Mq+1 so that we can apply (5.14) and (5.15). Theorem 44. A hypersequence converges if and only if σ σ q ∀q ∈ N ∃M ∈ Ne ∀n, m ∈ Ne≥M : m ≥ n ⇒ |xn − xm| < dρ . Proof. It suffices to apply the inequality |xn − xm| ≤ |xn − xn∨m| + |xn∨m − xm|. The second classical criterion for the existence of a hyperlimit is related to the notion of monotonic hypersequence. The existence of several chains in σNe does not allow to arrive at any M ∈ σNe starting from any other lower N ∈ σNe and using the successor operation only a finite number of times. For this reason, the following is the most natural notion of monotonic hypersequence: Definition 45. We say that (x ) is a non-decreasing (or increasing) hyper- n N∈σ Ne sequence if σ ∀n, m ∈ Ne : n ≥ m ⇒ xn ≥ xm. Similarly, we can define the notion of non-increasing (decreasing) hypersequence. σ ρ Theorem 46. Let (xn)n : Ne −→ Re be a non-decreasing hypersequence. Then ρ n σ o ∃ lim xn ⇐⇒ ∃ sup xn | n ∈ Ne , n∈σ Ne SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 23 n σ o and in that case they are equal. In particular, if xn | n ∈ Ne is complete from above for all the upper bounds, then ρ ρ σ ∃ lim xn ⇐⇒ ∃U ∈ Re ∀n ∈ Ne : xn ≤ U. n∈σ Ne Proof. Assume that (x ) converges to l and set S := {x | n ∈ σN}, we will n n∈σ Ne n e σ q show that l = sup(S). Now, using Def. 31, we have that ∀n ∈ Ne≥N : xn < l + dρ σ σ q for some N ∈ Ne. But from Def. 45 ∀n ∈ Ne : xn ≤ xn∨N < l + dρ . Therefore q σ xn ≤ l + dρ for all n ∈ Ne, and the conclusion xn ≤ l follows since q ∈ N is arbitrary. Finally, from Def. 31 of hyperlimit, for all q ∈ N we have the existence σ q of L ∈ Ne such that l − dρ < xL ∈ S which completes the necessity part of the proof. Now, assume that ∃ sup(S) =: l. We have to prove that ρ lim x = l. In n∈σ Ne n fact, using Rem. (i), we get q ∀q ∈ N ∃xN ∈ S : l − dρ < xN , q σ and xN ≤ xn ≤ l < l + dρ for all n ∈ Ne≥N by Def. 45 of monotonicity. That is, q |l − xn| = xn − l < dρ . 1/n Example 47. The hypersequence xn := dρ is non-decreasing. Assume that R (xn)n converges to l and that σ ≤ ρ for some R ∈ R>0. Since xn ≥ dρ, by Thm. 36, we get l ≥ dρ. Therefore, applying the logarithm and the exponential functions, from Thm. 37 we obtain that l = 1 because from σ ≤ ρR it follows that ρ lim log(dρ) = 0. But this is impossible since 1 ≈ 1 − dρ dρ1/n. Thereby, n∈σ Ne n n 1/n σ o @ sup dρ | n ∈ Ne>0 and this set is also not complete from above. 6. Limit superior and inferior We have two possibilities to define the notions of limit superior and inferior in a non-Archimedean setting such as ρRe: the first one is to assume that both σ σ αm := sup{xn | n ∈ Ne≥m} and inf{αm | m ∈ Ne} exist (the former for all m ∈ σNe); the second possibility is to use inequalities to avoid the use of supremum and infimum. In fact, in the real case we have ι ≤ supn≥m xn ≤ ι + ε if and only if ∀n ≥ m : xn ≤ ι + ε ∀ε ∃n¯ ≥ m : ι − ε ≤ xn¯ . σ ρ ρ Definition 48. Let (xn)n : Ne −→ Re be an hypersequence, then we say that ι ∈ Re is the limit superior of (xn)n if σ q (i) ∀q ∈ N ∃N ∈ Ne ∀n ≥ N : xn ≤ ι + dρ ; σ q (ii) ∀q ∈ N ∀N ∈ Ne ∃n¯ ≥ N : ι − dρ ≤ xn¯ . ρ Similarly, we say that σ ∈ Re is the limit inferior of (xn)n if σ q (iii) ∀q ∈ N ∃N ∈ Ne ∀n ≥ N : xn ≥ σ − dρ ; σ q (iv) ∀q ∈ N ∀N ∈ Ne ∃n¯ ≥ N : σ + dρ ≥ xn¯ . We have the following results (clearly, dual results hold for the limit inferior): σ ρ Theorem 49. Let (xn)n, (yn)n : Ne −→ Re be hypersequences, then (i) There exists at most one limit superior and at most one limit inferior. They are denoted with ρ lim sup x and ρ lim inf x . n∈σ Ne n n∈σ Ne n 24 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO n o (ii) If ∃ sup x | n ∈ σN =: α for all m ∈ σN, then ∃ ρ lim sup x if n e≥m m e n∈σ Ne n n σ o and only if ∃ inf αm | m ∈ Ne , and in that case ρ ρ n σ o lim sup xn = lim αm = inf αm | m ∈ Ne . σ n∈σ Ne m∈ Ne (iii) ρ lim sup (−x ) = −ρ lim inf x in the sense that if one of them exists, n∈σ Ne n n∈σ Ne n then also the other one exists and in that case they are equal. (iv) ∃ρ lim x if and only if ∃ρ lim sup x = ρ lim inf x . n∈σ Ne n n∈σ Ne n n∈σ Ne n (v) If ∃ρ lim sup x , ρ lim sup y , ρ lim sup (x + y ), then n∈σ Ne n n∈σ Ne n n∈σ Ne n n ρ ρ ρ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. n∈σ Ne n∈σ Ne n∈σ Ne σ In particular, if ∀N ∈ Ne ∀n,¯ nˆ ≥ N ∃n ≥ N : xn¯ + ynˆ ≤ xn + yn, then the existence of the single limit superiors implies the existence of the limit superior of the sum. (vi) If x , y ≥ 0 for all n ∈ σN and if ∃ρ lim sup x , ρ lim sup y , n n e n∈σ Ne n n∈σ Ne n ρ lim sup (x · y ), then n∈σ Ne n n ρ ρ ρ lim sup(xn · yn) ≤ lim sup xn · lim sup yn. n∈σ Ne n∈σ Ne n∈σ Ne σ In particular, if ∀N ∈ Ne ∀n,¯ nˆ ≥ N ∃n ≥ N : xn¯ · ynˆ ≤ xn · yn, then the existence of the single limit superiors implies the existence of the limit superior of the product. (vii) If ∃ ρ lim sup x =: ι, then there exists a sequence (¯n ) of σN such that n∈σ Ne n q q∈N e (a) n¯q+1 > n¯q for all q ∈ N; σ (b) limq→+∞ n¯q = +∞ in Re; (c) ∃ limq→+∞ xn¯q = ι. (viii) Assume to have a sequence (¯nq)q∈N satisfying the previous conditions (a), (b), (c) and σ ∀n ∈ Ne ∃p ∈ N :n ¯p ≥ n, xn ≤ xn¯p . (6.1) Then ∃ ρ lim sup x =: ι. n∈σ Ne n q Proof. (i): It suffices to use the inequalities |ι1 − ι2| ≤ |ι1 − xn¯ | + |xn¯ − ι2| ≤ 2dρ , wheren ¯ is taken from Def. 48.(ii). (ii): Lem. 24.(iii) implies that (αm)m is non-increasing. Therefore, we have n o ρ lim α = inf α | m ∈ σN if these terms exist from Thm. 46. But Cor. 29 n∈σ Ne m m e q q and Def. 48.(i) imply αm ≤ ι + dρ . Finally, Def. 48.(ii) yields ι − dρ ≤ xn¯ ≤ αm, which proves that ∃ ρ lim α = ρ lim sup x = ι. n∈σ Ne m n∈σ Ne m (iii): Directly from Def. 48. (iv): Assume that hyperlimit superior and inferior exist and are equal to l. From q q Def. 48.(i) and Def. 48.(iii) we get l − dρ ≤ xn ≤ l + dρ for all n ≥ N. Vice versa, q q assume that the hyperlimit exists and equals l, so that l −dρ ≤ xn ≤ l +dρ for all n ≥ N. Then both Def. 48.(i) and Def. 48.(iii) trivially hold. Finally, Def. 48.(ii) and Def. 48.(iv) hold taking e.g.n ¯ = N. SUP, INF AND HYPERLIMITS IN THE NON-ARCHIMEDEAN RING OF CGN 25 (v): Setting ρ ι := lim sup xn n∈σ Ne ρ j := lim sup yn n∈σ Ne ρ l := lim sup (xn + yn) , n∈σ Ne q q from Def. 48 we get l − dρ ≤ xn¯ + yn¯ ≤ ι + j + 2dρ , which implies l ≤ ι + j q for q → +∞. Adding Def. 48.(ii) we obtain ι + j − 2dρ ≤ xn¯ + ynˆ for somen ¯, σ nˆ ≥ N ∈ Ne. Therefore, if xn¯ +ynˆ ≤ xn +yn for some n ≥ N, this yields the second claim. Similarly, one can prove (vi). (vii): From Def. 48.(i), choose an Nq = N for each q ∈ N, i.e. σ q ∀q ∈ N ∃Nq ∈ Ne ∀n ≥ Nq : xn ≤ ι + dρ . (6.2) −q σ Applying Def. 48.(ii) with q > 0 and N = Nq ∨ (¯nq−1 + 1) ∨ [int(σε )] ∈ Ne, we q get the existence ofn ¯q ≥ Nq such that both (a) and (b) hold and ι − dρ ≤ xn¯q . Thereby, from (6.2) we also get (c). (viii): Write (c) as p p ∀q ∈ N ∃Qq ∈ N ∀p ∈ N≥Qq : ι − dρ ≤ xn¯p ≤ ι + dρ . (6.3) σ Set N :=n ¯Qq ∈ Ne. For n ≥ N, from (6.1) we get the existence of p ∈ N such that n¯p ≥ n and xn ≤ xn¯p . Thereby,n ¯p ≥ n¯Qq and hence p ≥ Qq because of (a) and q thus xn ≤ xn¯p ≤ ι + dρ . Finally, condition (ii) of Def. 48 follows from (6.3) and (b). It remains an open problem to show an example that proves as necessary the assumption of Thm. 49.(ii), i.e. that that the previous definition of limit superior and inferior is strictly more general than the simple transposition of the classical one. Example 50. (i) Directly from Def. 48, we have that ρ lim sup(−1)n = 1, ρ lim inf(−1)n = −1 σ n∈σ Ne n∈ Ne ρ c (ii) Let µ ∈ Re be such that µ|L = 1 and µ|Lc = −1, where L, L ⊆0 I. n q n¯ Then µ ≤ 1 and 1 − dρ ≤ µ if ni(¯n)ε is even for all ε small. There- fore ρ lim sup µn = 1, sup µn = 1, whereas ρ lim µn. n∈σ Ne n≥m @ n∈σ Ne (iii) From (vii) and (viii) of Thm. 49 it follows that for an increasing hypersequence (x ) , ∃ρ lim sup x if and only if ∃ρ lim x . Therefore, example 47 n n n∈σ Ne n n∈σ Ne n implies that ρ lim sup dρ1/n. @ n∈σ Ne 7. Conclusions In this work we showed how to deal with several deficiencies of the ring of Robinson-Colombeau generalized numbers ρRe: trichotomy law for the order re- lations ≤ and <, existence of supremum and infimum and limits of sequences with a topology generated by infinitesimal radii. In each case, we obtain a faithful gen- eralization of the classical case of real numbers. We think that some of the ideas 26 A. MUKHAMMADIEV, D. TIWARI, G. APAABOAH, AND P. GIORDANO we presented in this article can inspire similar works in other non-Archimedean set- tings such as (constructive) nonstandard analysis, p-adic analysis, the Levi-Civita field, surreal numbers, etc. Clearly, the notions introduced here open the possi- bility to extend classical proofs in dealing with series, analytic generalized func- tions, sigma-additivity in integration of generalized functions, non-Archimedean functional analysis, just to mention a few. Acknowledgments: The authors would like to thank L. 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[23] Scarpal´ezos, D., Colombeau’s generalized functions: topological structures; microlocal prop- erties. A simplified point of view, I. Bull. Cl. Sci. Math. Nat. Sci. Math. no. 25 (2000), 89–114. [24] Shamseddine, K., A brief survey of the study of power series and analytic functions on the Levi-Civita fields, Contemporary Mathematics, Volume596, 2013. [25] Vernaeve, H., Generalized analytic functions on generalized domains, arXiv:0811.1521v1, 2008. [26] Vernaeve, H., Ideals in the ring of Colombeau generalized numbers. Comm. Alg., 38 (2010), no. 6, 2199–2228. University of Vienna, Austria Email address: [email protected] University of Vienna, Austria Email address: [email protected] University Grenoble Alpes, France Email address: [email protected] Wolfgang Pauli Institute, Vienna, Austria Email address: [email protected]s y (iv) x ≤ y ⇒ x s y or x 6>s y. In the second case, Lem.5 implies x ≤ y. If y ≤ x then x = y; otherwise, once again by Lem.5, we get x q1. Applying again (4.1) q2 we get σ − dρ < s¯2 for somes ¯2 ∈ S \{s¯1}. Recursively, this process proves that S is infinite. On the other hand, if S = {s1, . . . , sn} and si = [siε], then sup ([{s1ε, . . . , snε}]) = s1 ∨...∨sn. In fact, s1 ∨...∨sn = [maxi=1,...,n snε] ∈ [{s1ε, . . . , snε}]. Theorem 22. There is at most one sharp supremum of S, which is denoted by sup(S).