A Construction of the Real Numbers
In Chapter 1 we have described the sets N, Z,andQ of natural numbers, integers, and rational numbers, respectively, as follows:
N = {1, 2, 3,...}, Z = {...,−3, −2, −1, 0, 1, 2, 3,...}, and m Q = : m, n ∈ Z,n=0 . n The set R of “real numbers” was then introduced as a set containing Q and the “irrational numbers” in such a way that the elements of R are in one- to-one correspondence with the points on the “number line”. But R defies a simplistic description such as that given above for N, Z,andQ. Thus, while we can conceive easily what a rational number is, it is a little harder to say precisely what a real number is. For this reason, we made an assumption that there exists a set R containing Q that satisfies the three sets of properties given in Section 1.1, namely the Algebraic Properties A1-A5, the Order Properties O1-O2, and the Completeness Property. The main aim of this appendix is to show that such a set R does indeed exist and is essentially unique. The approach that we shall take is due to Cantor, and uses Cauchy sequences of rational numbers. In what follows, we shall assume familiarity with the set Q and the usual algebraic operations on Q as well as the usual order relation that permits us to talk of the subset Q+ of positive rational numbers in such a way that the properties A1-A5 and O1-O2 in Section 1.1 are satisfied if we replace R by Q throughout. This appendix is divided into three sections, which are organized as follows. In the first section below, we discuss some preliminaries about equivalence relations and equivalence classes. Then in the next section, we outline a construction of R using Cauchy sequences of rational numbers. The “uniqueness” of R is formally established in the last section.
A.1 Equivalence Relations
The notion of an equivalence relation is basic to much of mathematics, and it will be useful in our formal construction of R from Q. The most basic equivalence relation on any set is that of equality denoted by =. Fundamental properties of this relation motivate the following definition.
© Springer Nature Switzerland AG 2018 503 S. R. Ghorpade, B. V. Limaye, A Course in Calculus and Real Analysis, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-030-01400-1 504 A Construction of the Real Numbers
Let S be a set. A relation on S is a subset of S × S.If∼ is a relation on S and a, b ∈ S, then we usually write a ∼ b to indicate that the ordered pair (a, b) is an element of the subset ∼ of S × S.Arelation∼ on S is called an equivalence relation if (i) ∼ is reflexive,thatis,a ∼ a for all a ∈ S, (ii) ∼ is symmetric,thatis,b ∼ a whenever a, b ∈ S satisfy a ∼ b, and (iii) ∼ is transitive,thatis,a ∼ c whenever a, b, c ∈ S satisfy a ∼ b and b ∼ c. If ∼ is an equivalence relation on a set S and if a ∈ S, then the set {x ∈ S : x ∼ a} is called the equivalence class of a and is denoted1 by [a]; in general, a subset E of S is called an equivalence class (with respect to ∼)ifE =[a]forsomea ∈ S. A key fact about equivalence relations is the following result, which basically says that an equivalence relation on a set partitions the set into disjoint equivalence classes.
Proposition A.1. Let S beasetandlet∼ be an equivalence relation on S. Then any two equivalence classes (with respect to ∼) are either disjoint or identical. Consequently, if E denotes the collection of distinct equivalence classes with respect to ∼,then ( S = E, E∈E where the union is disjoint.
Proof. Let a, b ∈ S and suppose the equivalence classes [a]and[b]arenot disjoint, that is, there exists c ∈ [a] ∩ [b]. Then c ∼ a and c ∼ b. Hence using the fact that ∼ is an equivalence relation, we see that for every x ∈ S,
x ∈ [a] ⇐⇒ x ∼ a ⇐⇒ x ∼ c ⇐⇒ x ∼ b ⇐⇒ x ∈ [b].
This shows that [a]=[b]. Thus any two equivalence classes are either disjoint ∈ ∈ or identical. Finally, since a [a]foreacha S,weobtainS = a∈S[a]. We give several examples of equivalence relations and corresponding equiv- alence classes below. The detailed verification of the assertions made in these examples is left to the reader.
Examples A.2. (i) On the set N, define a relation ∼ by
m ∼ n ⇐⇒ m and n have the same parity,thatis,(−1)m =(−1)n.
Then ∼ is an equivalence relation. There are exactly two equivalence classes with respect to ∼, namely the set of odd positive integers and the set of even positive integers.
1 When S ⊆ Q, the notation [a] for the equivalence class of an element a of S conflicts with the notation used in the text for the integer part of a.Toavoidany possible confusion, we shall always use in this appendix the notation a for the integer part of a. A.2 Cauchy Sequences of Rational Numbers 505
(ii) On the set N × N, define a relation ∼ by
(a, b) ∼ (c, d) ⇐⇒ a + d = b + c for (a, b), (c, d) ∈ N × N.
Then ∼ is an equivalence relation, and the equivalence classes with respect to ∼ are in one-to-one correspondence with the set Z of all integers. (iii) Let S = {(m, n):m, n ∈ Z and n =0 }.Therelation∼ on S defined by
(a, b) ∼ (c, d) ⇐⇒ ad = bc for (a, b), (c, d) ∈ S
is an equivalence relation on S, and the equivalence classes with respect to ∼ are in one-to-one correspondence with the set Q of all rational numbers. (iv) Let n ∈ N. Consider the relation ∼ on Z defined by
a ∼ b ⇐⇒ a − b is divisible by n for a, b ∈ Z.
Then ∼ is an equivalence relation, called congruence modulo n.There are exactly n distinct equivalence classes with respect to ∼ given by C0,C1,...,Cn−1,wherefor0≤ i We remark that examples (ii) and (iii) above can be used to formally construct Z from N, and to construct Q from Z. For an axiomatic treatment of N, we refer to the book of Landau [54]. A.2 Cauchy Sequences of Rational Numbers We shall now define the notion of a Cauchy sequence in Q. This is completely analogous to the notion discussed in Chapter 2, except that we will refrain from using real numbers anywhere. In particular, will denote a positive ratio- nal number, that is, ∈ Q+. Note that since it is well understood what positive rational numbers are, the notion of the absolute value of a rational number is well-defined and satisfies the basic properties given in Proposition 1.8. A sequence in Q is a function from N to Q. We usually write (an)to denote the sequence a : N → Q defined by a(n):=an for n ∈ N. The rational number an is called the nth term of the sequence (an). A sequence (an)of rational numbers is said to be 1. bounded above if there exists α ∈ Q such that an ≤ α for all n ∈ N, 2. bounded below if there exists β ∈ Q such that an ≥ β for all n ∈ N, 3. bounded if it is bounded above as well as bounded below, + 4. Cauchy if for every ∈ Q , there exists n0 ∈ N such that |an − am| < for all m, n ∈ N with m, n ≥ n0. 506 A Construction of the Real Numbers Weshallalsosaythatasequence(cn) of rational numbers is null,andwrite + cn → 0, if for every ∈ Q ,thereisn0 ∈ N such that |cn| <for all n ≥ n0. Examples A.3. (i) Let (an) be the sequence in Q defined by an := 1/n + for n ∈ N.Then(an) is a null sequence. Indeed, given any ∈ Q ,say = p/q,wherep, q ∈ N, the positive integer n0 := q + 1 satisfies n0 > 1/. Hence |an| <for all n ≥ n0. (ii) Let (an) be the sequence in Q defined by an := (n − 1)/n for n ∈ N.Then + (an) is a Cauchy sequence. To see this, let = p/q ∈ Q be given, where p, q ∈ N. Now the positive integer n0 := 2(q +1)satisfies n0 > 2/. Hence n − 1 m − 1 1 1 2 |an − am| = − = − ≤ < for all m, n ≥ n0. n m m n n0 Note that although (an) is a Cauchy sequence, it is not a null sequence. In fact, |an|≥1/2 for all n ≥ 2. Proposition A.4. (i) Every Cauchy sequence of rational numbers is bounded. (ii) Every null sequence of rational numbers is Cauchy. (iii) Let (an) be a Cauchy sequence of rational numbers that is not a null ∈ Q+ ∈ N | − | sequence. Then there exist 0 and n0 such that an an0 <0 and |an|≥0 for all n ≥ n0. Proof. (i) Let (an) be a Cauchy sequence. Then there exists k ∈ N such that |an − am| < 1 for all m, n ≥ k. Consequently, |an|≤α for all n ∈ N, where α := max {|a1|, ..., |ak−1|, |ak| +1} . Hence (an) is bounded. + (ii) Let (cn) be a null sequence. Given any ∈ Q , there exists n0 ∈ N such that |cn| </2 for all n ≥ n0.Then |c − c |≤|c | + |c | < + = for all n ≥ n . n m n m 2 2 0 Hence (cn)isCauchy. + (iii) Since (an) is not a null sequence, there exists ∈ Q such that for ∈ N ≥ | |≥ every k , there exists n1 k satisfying an1 .Also,since(an)is a Cauchy sequence, there exists n0 ∈ N such that |am − an| </2 for all ≥ ≥ | |≥ m, n n0.Letk := n0, and find n1 n0 such that an1 .Then ≤|a |≤|a − a | + |a |≤ + |a |, and hence |a |≥ for all n ≥ n . n1 n1 n n 2 n n 2 0 + Thus 0 := /2 ∈ Q and n0 ∈ N have the desired property. Now let us define C := the set of all Cauchy sequences of rational numbers. Further, consider the relation ∼ on C defined by (an) ∼ (bn) ⇐⇒ (an − bn) is a null sequence, where (an), (bn) ∈ C . A.2 Cauchy Sequences of Rational Numbers 507 Proposition A.5. The relation ∼ is an equivalence relation on C . Proof. Clearly, ∼ is reflexive and symmetric. Suppose (an), (bn), (cn) ∈ C are + such that (an) ∼ (bn)and(bn) ∼ (cn). Given any ∈ Q ,thenumber/2is + also in Q . Hence there exist n1,n2 ∈ N such that |a − b | < for all n ≥ n and |b − c | < for all n ≥ n . n n 2 1 n n 2 2 Now if n0 =max{n1,n2},thenforeachn ≥ n0, |a − c | = |(a − b )+(b − c )|≤|a − b | + |b − c | < + = . n n n n n n n n n n 2 2 This shows that (an) ∼ (cn). Thus ∼ is transitive as well. We are now ready to define a model for R that we seek to construct. Let R := the set of all equivalence classes of elements of C with respect to ∼. As in the previous section, the equivalence class of a Cauchy sequence (an) in C with respect to ∼ will be denoted by [(an)]. Given any r ∈ Q, the constant sequence (r), that is, the sequence (rn)withrn = r for all n ∈ N, is clearly Cauchy. We will denote by Q the subset of R consisting of the equivalence classes of constant sequences of rational numbers. It is clear that the map from Q to Q given by r −→ [(r)] is one-one and onto. Thus we can, and will, identify Q with Q. In particular, the equivalence classes of the constant sequences (0) and (1) will be denoted simply by 0 and 1, respectively. We now define addition and multiplication on the set R as follows. [(an)] + [(bn)] = [(an + bn)] and [(an)] · [(bn)] = [(anbn)] for (an), (bn) ∈ C . Proposition A.6. The operations of addition and multiplication on R are well-defined and satisfy the following algebraic properties: A1. a +(b + c)=(a + b)+c and a(bc)=(ab)c for all a, b, c ∈R. A2. a + b = b + a and ab = ba for all a, b ∈R. A3. a +0=a and a · 1=a for all a ∈R. A4. Let a ∈R. Then there exists a ∈Rsuch that a + a =0.Further,if a =0 , then there exists a∗ ∈Rsuch that aa∗ =1. A5. a(b + c)=ab + ac for all a, b, c ∈R. Proof. To show that the operations of addition and multiplication on R are ∈ C well-defined, it suffices to show that for all (an), (an), (bn), (bn) , ∼ ∼ ⇒ ∼ ∼ (an) (an)and(bn) (bn)= (an + bn) (an + bn)and(anbn) (anbn). ∼ The assertion (an + bn) (an + bn) follows from the definition, since | − |≤| − | | − | ∼ (an + bn) (an + bn) an an + bn bn . To see that (anbn) (anbn), 508 A Construction of the Real Numbers ∈ Q+ | |≤ we use part (i) of Proposition A.4 and obtain α ,β such that an α and |bn|≤β for all n ∈ N,sothat | − | | − − |≤ | − | | − | anbn anbn = (an an)bn + an(bn bn) β an an + α bn bn . − → − → − → Now since an an 0andbn bn 0, it is readily seen that anbn anbn 0. Having established that addition and multiplication on R are well-defined, we see that properties A1, A2, A3, and A5 are immediate consequences of the definition and the corresponding properties of rational numbers. Moreover, for every a =[(an)] ∈R, the element a := [(−an)] is clearly in R and it satisfies a + a = 0. Finally, suppose a =[(an)] ∈Ris such that + a = 0. Then by part (iii) of Proposition A.4, there exist 0 ∈ Q and n0 ∈ N such that |an|≥0 for all n ≥ n0. In particular, an = 0 for all n ≥ n0. ∗ Q ∗ ≤ ∗ Define the sequence (an)in by an := 1 for 1 n Now let us turn to order properties. We shall say that a sequence (an) ∈ C is positive if it satisfies the following property: + There exist r ∈ Q and n0 ∈ N such that an ≥ r for all n ≥ n0. ∈ C ∼ Note that if (an) is such that (an) (an)and(an) satisfies the above ∼ ∈ N property, then so does (an). Indeed, since (an) (an), there exists n1 | − | ≥ such that an an Proposition A.7. The set R+ satisfies the following order properties: O1. Given any a ∈R, exactly one of the following statements is true: a ∈R+; a =0; −a ∈R+. A.2 Cauchy Sequences of Rational Numbers 509 O2. If a, b ∈R+,thena + b ∈R+ and ab ∈R+. Proof. Let (an) ∈ C be such that [(an)] = 0. By part (iii) of Proposition A.4, ∈ Q+ ∈ N | − | | |≥ there exist 0 and n0 such that an an0 <0 and an 0 for ≥ all n n0. In particular, an0 =0.Nowifan0 > 0, then the above inequalities − − ≥ imply an = an0 +(an an0 ) >0 0 = 0 for all n n0, and consequently, ≥ ≥ − ≥ ≥ an 0 for all n n0. Likewise, if an0 < 0, then an 0 for all n n0. + + Thus [(an)] ∈R or −[(an)] = [(−an)] ∈R .ThisprovesO1. + Next,if(an), (bn) ∈ C are such that [(an)], [(bn)] ∈R , then there exist + r1,r2 ∈ Q and n1,n2 ∈ N such that an >r1 for all n ≥ n1 and bn >r2 for all n ≥ n2. Now if we let n0 =max{n1,n2}, then we clearly have an + bn >r1 + r2 for all n ≥ n0 and anbn >r1r2 for all n ≥ n0. + + + Since r1 +r2,r1r2 ∈ Q ,weobtain[(an)]+[(bn)] ∈R and [(an)][(bn)] ∈R . This proves O2. Using the set R+, we can define an order relation on R exactly as in Section 1.1, namely, for all a, b ∈R,wewritea Proposition A.8. Given any a ∈R,thereissomek ∈ N such that k>a. Proof. First note that if a ∈Q,thena corresponds to a unique rational number p/q,wherep ∈ Z and q ∈ N. Hence k := |p| + 1 clearly satisfies k>a. Now suppose a =[(an)] is an arbitrary element of R,where(an)isa Cauchy sequence of rational numbers. Then for =1/2, there is n0 ∈ N such 510 A Construction of the Real Numbers that |an − am| < 1/2 for all n, m ≥ n0. Since the archimedean property holds ∈ N | | for rational numbers, there exists such that an0 <. Hence for each n ≥ n0,weobtain 1 1 |a |≤|a −a |+|a | < + and hence |(1+)−a |≥(1+)−|a | > . n n n0 n0 2 n n 2 It follows that [(an)] < [(1 + )], that is, k := 1 + ∈ N satisfies k>a. The archimedean property proved in Proposition A.8 enables us to define the integer part of every x ∈Rexactly as in the paragraph following Propo- sition 1.3 of Chapter 1, and this, in turn, permits us to deduce that between any two elements of R, there is a rational number. It is important to note that this result uses only the algebraic and order properties together with the archimedean property (Propositions A.6, A.7, and A.8). Proposition A.9. Given any a, b ∈Rwith a Proof. The proof is identical to that of Proposition 1.6 and hence omitted. A more general result (Proposition A.12) is proved in the next section. Corollary A.10. Let (rn) ∈ C and let a =[(rn)] be the corresponding ele- ment of R.Ifthereexistsn0 ∈ N such that rn ≥ 0 for all n ≥ n0,thena ≥ 0. More generally, if there exist α, β ∈Rand n0 ∈ N such that β ≤ rn ≤ α for all n ≥ n0,thenβ ≤ a ≤ α. Proof. Suppose there exists n0 ∈ N such that rn ≥ 0 for all n ≥ n0. Let, if possible, a<0. Then by Proposition A.9, there exists s ∈ Q such that a Proposition A.11. Every nonempty subset of R that is bounded above has a supremum. Proof. Let S be a nonempty subset of R that is bounded above. Since S is nonempty, there is some a0 ∈S, and since S is bounded above, there is some α0 ∈Rsuch that α0 is an upper bound of S,thatis,a ≤ α0 for all a ∈S. Now let β1 := (a0 + α0)/2. If β1 is an upper bound of S,weleta1 := a0 and α1 := β1,whereasifβ1 is not an upper bound of S, then there exists b ∈S A.2 Cauchy Sequences of Rational Numbers 511 such that β1 Next, we replace (a0,α0)by(a1,α1) and proceed as before. In general, given i n ∈ N and ai ∈Sand upper bounds αi of S with 0 ≤ (αi − ai) ≤ (α0 − a0)/2 for 0 ≤ i ≤ n − 1andwitha0 ≤ a1 ≤··· ≤an−1 and α0 ≥ α1 ≥··· ≥αn−1, we choose an ∈Sand an upper bound αn of S as follows. Let βn := (an−1 + αn−1)/2. If βn is an upper bound of S,weletan := an−1 and αn := βn, whereas if βn is not an upper bound of S, then there exists b ∈Ssuch that βn Note that if an = αn for some n ≥ 0, then clearly αn is the supremum of S. Now suppose an <αn for all n ≥ 0. By Proposition A.9, for each n ∈ N, there exists rn ∈ Q such that an rn − rm <αn − rm <αn − am ≤ αn − an. In a similar way, since αm ≤ αn,weobtain rn − rm >an − rm >an − αm ≥ an − αn. It follows that α − a α − a |r − r | <α − a ≤ 0 0 ≤ 0 0 < for all m, n ≥ k. n m n n 2n 2k Thus (rn) ∈ C ,andsoα := [(rn)] ∈R. We shall now show that α is the supremum of S. To this end, let us first observe that for every fixed m ∈ N, the inequalities am ≤ an Hence αm A.3 Uniqueness of a Complete Ordered Field In the previous section, we have shown that the set R possesses all the prop- erties that were postulated for R in Chapter 1. In other words, we have es- tablished the existence of the set of all real numbers. We will now prove its “uniqueness”. First, we introduce some useful terminology. By a field we shall mean a set F that has operations of addition and multiplication (that is, maps from F × F to F that associate elements a + b and ab of F to (a, b) ∈ F × F ) and has distinct elements 0F and 1F in it such that the five algebraic properties A1–A5 in Proposition A.6 are satisfied with R replaced throughout by F , and with 0 and 1 replaced by 0F and 1F .Itis easy to see that in a field F ,elements0F and 1F satisfying A3 are unique, and these are sometimes called the additive identity and the multiplicative identity of F , respectively. Note that Q and R are examples of fields. A special case of Example A.2 (iv), namely the set Z/pZ of residue classes modulo a prime number p, is a field having only finitely many elements. If a field F contains a subset F + satisfying the two order properties O1– O2 with R+ replaced throughout by F +,thenF is called an ordered field; in this case, for every a, b ∈ F ,wewritea (r + s)F = rF + sF , (rs)F = rF sF , and r Thus F contains a copy of Q,namelyQF := {rF : r ∈ Q}. In an ordered field F ,theabsolute value of an element can be defined as in the case of Q or R. Thus for every a ∈ F ,welet|a| := a if a ≥ 0F and |a| := −a if a<0F .Itis easily seen that |ab| = |a||b| and |a + b|≤|a| + |b| for all a, b ∈ F . Let F be an ordered field. We say that F is archimedean if for every a ∈ F , there exists n ∈ N such that nF >a. and we say that F is complete if every nonempty subset of F that is bounded above has a supremum in F . For example, both Q and R are archimedean ordered fields, and R is complete (thanks to Proposition A.11), but Q is not. In general, by arguing as in the proof of Proposition 1.3, we readily see that a complete ordered field is archimedean. Let F be an archimedean ordered field and let a ∈ F . Then there exist m, n ∈ N such that mF > −a and nF >a,thatis,−mF Proposition A.12. Let F be an archimedean ordered field and let a1,a2 ∈ F satisfy a1 −1 Proof. Since F is archimedean, there exists n ∈ N such that nF > (a2 −a1) , −1 that is, (nF ) < (a2 −a1). Let m ∈ N be such that mF = nF a1 +1F .Then mF − 1F ≤ nF a1 −1 −1 −1 a1 Thus r = m/n ∈ Q satisfies a1 Corollary A.13. Let F be an archimedean ordered field. Suppose a ∈ F sat- + isfies |a| <F , that is, −F Proof. In case a>0F , by Proposition A.12, there exists r ∈ Q such that 0F Let K and F be ordered fields. A map f : K → F is called an order isomorphism if f is both one-one and onto, and f preserves algebraic and order operations, that is, for all x, y ∈ K, f(x + y)=f(x)+f(y),f(xy)=f(x)f(y), and x If such a map exists, then we say that F is order isomorphic to K. Proposition A.14. Every complete ordered field is order isomorphic to R. 514 A Construction of the Real Numbers Proof. Let F be a complete ordered field. Define f : R→F by f(x):=supFx, where Fx := {rF : r ∈ Q and r ≤ x} for x ∈R. Note that f is well-defined. Indeed, given any x ∈R, by Proposition A.9, there exist s, t ∈ Q such that x − 1 F F f(x + y) − f(x) − f(y) < (sF − rF )+(vF − uF ) < + = F . 2F 2F By a similar argument, f(x+y)−f(x)−f(y) > −F . Now by Corollary A.13, we obtain f(x + y)=f(x)+f(y). Thus, f preserves addition. A.3 Uniqueness of a Complete Ordered Field 515 To show that f preserves multiplication, first consider x, y ∈Rwith x>0 and y>0. Let ∈ Q+ be given. By Proposition A.8, there exists n ∈ N such that n>x, n>y,andn>/6. Now using Proposition A.9 and the assumption that x>0andy>0, we obtain r, s, u, v ∈ Q+ such that x − F F sF vF − F F sF vF − It follows that −F Remark A.16. In this book, we have used the word completeness (of R,or more generally, of any ordered field F ) to mean that every nonempty subset that is bounded above has a supremum. This is sometimes referred to as order completeness, especially when contrasted with other notions such as monotone completeness and Cauchy completeness that are defined as follows. An ordered field F is said to be monotone complete (resp. Cauchy complete) if every monotonic bounded sequence (resp. Cauchy sequence) in F is convergent. Note that in an ordered field, the notions of absolute value and of a sequence being monotonic, bounded, convergent, or Cauchy are defined in the same way as in the case of R. Arguing as in the proofs of Propositions 2.8 and 2.22, we readily see that for every ordered field F , F is complete =⇒ F is monotone complete =⇒ F is Cauchy complete. It can be shown that Cauchy completeness implies (order) completeness, pro- vided that the ordered field is archimedean. (Compare Exercise 2.42 of Chap- ter 2.) Thus for an archimedean ordered field, the three notions of complete- ness are equivalent. Moreover, an ordered field that is monotone complete is necessarily archimedean (because otherwise, the sequence (nF )wouldbe convergent, and if nF converges to a, then there exists n0 ∈ N such that a−1F Although the set R of all real numbers is “complete”, it has a lacuna from an algebraic point of view. Namely, there are polynomials with real coefficients that have no root in R. The simplest among these is the polynomial x2 +1. By adjoining to R an “imaginary” root i of x2+1, we obtain the complex numbers, which are sometimes “defined” as the numbers of the form x + iy,where x, y ∈ R. It is not difficult to give a formal and precise definition of complex numbers and in particular, of the number i. We do this in Section B.1, and then outline how several of the notions about real-valued functions can be extended to complex-valued functions. Next, in Section B.2, we prove a remarkable result known as the Fundamental Theorem of Algebra, which basically says that if one can solve x2 + 1 = 0, then one can solve every polynomial equation in one variable with real or complex coefficients! B.1 Complex Numbers and Complex Functions A complex number is defined as an ordered pair of real numbers. The set of all complex numbers is denoted by C. Addition and multiplication in C are defined as follows. For all (x1,y1), (x2,y2) ∈ C,let (x1,y1)+(x2,y2):=(x1 + x2,y1 + y2), and (x1,y1)(x2,y2):=(x1x2 − y1y2,x1y2 + y1x2). It is easily seen that with respect to these algebraic operations, C is a field. Moreover, the map given by x −→ (x, 0) gives a one-one map from R to C, which preserves the algebraic operations. With this in view, we regard R as a subset of C by identifiyng a real number x with the ordered pair (x, 0) in C. We define i := (0, 1). With the identification of R with a subset of C as above, we can write any (x, y) ∈ C as x + iy.Notethati2 = −1, where we have again identified −1with(−1, 0). Let z ∈ C. As noted above, z = x + iy for unique x, y ∈ R.Wecallx the real part of z and denote it by (z), and we call y the imaginary part of z and denote it by (z). The complex number x − iy is called the conjugate of z = x + iy and is denoted by z.Wealso define the absolute√ value or the modulus of z to be the nonnegative real number |z| := zz = x2 + y2. Note that this definition is consistent with © Springer Nature Switzerland AG 2018 517 S. R. Ghorpade, B. V. Limaye, A Course in Calculus and Real Analysis, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-030-01400-1 518 B Fundamental Theorem of Algebra that of the absolute value of a real number, and that the Triangle Inequality |z1 + z2|≤|z1| + |z2| holds for all z1,z2 ∈ C. Also, note that max {| (z)|, | (z)|} ≤ |z|≤| (z)| + | (z)| for all z ∈ C. Observe that C is not an ordered field. Indeed, if C had a subset C+ satisfying the two order properties O1 and O2 (as in Proposition A.7 with R+ replaced by C+ and R replaced by C), then by O1, either i ∈ C+ or −i ∈ C+.NowO2 implies that −1=(±i)2 ∈ C+ and 1 = (−1)2 ∈ C+. This contradicts O1. The complex exponential of a complex number z is defined by ez := ex(cos y + i sin y), where x = (z)andy = (z). Note that |ez| = e(z) and therefore ez = 0 for all z ∈ C.Notealsothat eiθ =cosθ + i sin θ for all θ ∈ R. In particular, eiπ +1=0. We now consider complex-valued functions of a real variable, that is, functions whose codomain is C and whose domain is a subset of R. For exam- ple, h : R → C defined by h(t):=eit is a complex-valued function of a real variable. In general, if D ⊆ R and if h : D → C is a complex-valued function of a real variable, then there are unique functions f,g : D → R such that h(t)=f(t)+ig(t)fort ∈ D.Wecallf the real part and g the imaginary part of h,andwriteh = f + ig. The notions of boundedness, continuity, dif- ferentiability and Riemann integration extend to complex-valued functions of a real variable in a straightforward manner. In fact, we define h : D → C to be • bounded on D if both f and g are bounded on D, • continuous at a point c of D if both f and g are continuous at c, • differentiable at an interior point c of D if both f and g are differentiable at c; in this case, the complex number h(c):=f (c)+ig(c) is called the derivative of h at c, In case D := [a, b]forsomea, b ∈ R with a ≤ b,andh : D → C is bounded, then we define h to be integrable on [a, b']ifbothf 'and g are integrable [a, b], and in this case, the complex number b f(t)dt + i b g(t)dt is called the ' a a b integral of h over [a, b] and denoted by a h(t)dt. For example, if D =[0,π]andh : D → R is defined by h(t):=eit,thenh is differentiable at each t ∈ D and h(t)=ieit.Also,h is integrable and & & & b π π 1 h(t)dt = cos tdt + i sin tdt =0− i((−1) − 1) = 2i = eiπ − e0 . a 0 0 i In a similar way, we can consider complex-valued functions of two or more real variables. Thus, for example, if D ⊆ R2 and h : D → C is a complex- valued function of two real variables, then there are unique real-valued func- tions f,g : D → R such that h(t, u)=f(t, u)+ig(t, u)for(t, u) ∈ D.Asin Section 10.6, we can fix one variable and differentiate or integrate with respect to another. Thus we let B.2 Polynomials and Their Roots 519 D1h(t, u)=D1f(t, u)+iD1g(t, u)andD2h(t, u)=D2f(t, u)+iD2g(t, u), where D1 indicates differentiation with respect to the first variable t (treating u as a constant) and D2 indicates differentiation with respect to the second variable u (treating t as a constant). Likewise, if D ⊆ R2 is of the form D =[a, b] × E for some a, b ∈ R with a ≤ b and E ⊆ R,andifh : D → C is continuous in t (which means both f and g are continuous in t), then we can consider the integral function H : E → C given by & & & b b b H(u)= h(t, u)dt := f(t, u)dt + i g(t, u)dt for u ∈ E. a a a Moreover, the following analogue of Proposition 10.52 holds. Proposition B.1. Let E be an interval in R,andleta, b ∈ R with a ≤ b. Suppose h :[a, b] × E → C is a complex-valued function of two real variables × ∈ such that' D2h exists and is bounded on [a, b] E, and for each u E,the integral b D h(t, u)dt exists. Then the integral function H : E → C given by 'a 2 ' b b ∈ H(u):= a h(t, u)dt is differentiable and H (u)= a D2h(t, u)dt for u E. Proof. Use Proposition 10.52 for the real and imaginary parts of h. B.2 Polynomials and Their Roots In this section, we shall give a proof due to Paul Loya [60] of the Fundamen- tal Theorem of Algebra. The statement involves polynomials (in one variable) with complex coefficients, which are defined in the same way as polynomi- als with real coefficients. (See Section 1.3.) Notions of the degree and of the leading coefficient of a nonzero polynomial are also defined in the same way. In particular, by a nonconstant polynomial we mean a polynomial of pos- itive degree. Also, by a monic polynomial we mean a polynomial whose leading coefficient is 1. We will begin by proving an elementary property of polynomials that will be useful later. Lemma B.2. Let p(z) be a polynomial with coefficients in C. Then for every z0 ∈ C,thereexistsδ0 > 0 such that |p(z)|≥|p(z0)|/2 for all z ∈ C with |z − z0| <δ0. d d−1 Proof. Write p(z)=cdz + cd−1z + ···+ c1z + c0,wherec0,c1,...,cd ∈ C. Let us first consider z0 := 0, so that p(z0)=c0.Notethatifc0 =0,then there is nothing to prove. Let us now assume that c0 =0.Bywriting c0 = − d−1 d−2 ··· p(z) z cdz + cd−1z + + c1 , we see that the triangle inequality d−1 gives |p(z)|≥|c0|−|z| |cd||z| + ···+ |c1| . Consider δ0 ∈ R defined by δ0 := |c0|/2(|cd| + ···+ |c1| + |c0|). Then 0 <δ0 ≤ 1/2 < 1and 520 B Fundamental Theorem of Algebra |c | |z| <δ =⇒|z| |c ||z|d−1 + ···+ |c | ≤ δ (|c | + ···+ |c |) < 0 . 0 d 1 0 d 1 2 Consequently, |p(z)|≥|c0|/2=|p(0)|/2 whenever |z| <δ0. Next, consider any z0 ∈ C. Substituting z =(z − z0)+z0 in p(z), we see ∗ − ∗ ∗ d ··· ∗ ∗ that p(z)=p (z z0), where p (w)=cdw + + c1w + c0 is a polynomial in ∗ ∗ ∈ C w whose coefficients c0,...,cd are determined by c0,...,cd and z0,and ∗ moreover, c0 = p(z0). From the first part of the proof, we find δ0 > 0such | ∗ |≥|∗| ∈ C | | − that p (w) c0 /2 for all w with w <δ0.Putingw = z z0,we obtain the desired result. We remark that the above lemma can also be deduced from the continuity of polynomial functions of a complex variable. However, since we have not discussed continuity of functions of a complex variable, we have chosen to give a direct proof. Proposition B.3 (Fundamental Theorem of Algebra). Every noncon- stant polynomial with coefficients in C has a root in C. Proof. Let p(z) be a nonconstant polynomial of degree d with coefficients in C. Dividing p(z) by its leading coefficient, we may assume that p(z)ismonic. d d−1 Thus we can write p(z)=z +cd−1z +···+c1z+c0,wherec0,...,cd−1 ∈ C. Suppose p(z) has no root in C,thatis,p(z0) = 0 for all z0 ∈ C. In particular, c0 = p(0) = 0. Consider the function h :[−π, π] × R → C defined by 1 1 ∈ − ∈ R h(t, u):= it = d idt it for t [ π, π]andu . p (ue ) u e + ···+ c1ue + c0 Clearly, h is differentiable in each of the two variables t and u.Moreover,a direct computation shows that d−1 idt it − du e + ···+ c1e D2h(t, u)= for t ∈ [−π, π]andu ∈ R. d idt it 2 (u e + ···+ c1ue + c0) It is clear that for each fixed u ∈ R, the function from [−π, π]toC given by t −→ D2h(t, u) is continuous. Now let α ∈ R with α>0. Then d−1 idt it d−1 du e + ···+ c1e ≤ dα + ···+ |c1| for t ∈ [−π, π]andu ∈ [−α, α]. Next, we show that there exists δ>0 such that p ueit ≥ δ for all t ∈ [−π, π] and u ∈ [−α, α]. Suppose this is not the case. Then there are sequences (tn)in itn [−π, π]and(un)in[−α, α] such that p une < 1/n for all n ∈ N.Bythe Bolzano–Weierstrass Theorem (Proposition 2.17), there exist a subsequence ∈ − → (tnk )of(tn)andt0 [ π, π] such that tnk t0. In turn, there exist a subsequence (u )of(u )andu ∈ [−α, α] such that u → u .Let nkj nk 0 nkj 0 itn z := u e kj for j ∈ N and let z := u eit0 .Thenp(z ) =0,andby j nkj 0 0 0 Lemma B.2, there exists δ0 > 0 such that |p(z)|≥|p(z0)|/2 for all z ∈ C with B.2 Polynomials and Their Roots 521 |z − z0| <δ0.Sincezj → z0, there exists j0 ∈ N such that |zj − z0| <δ0 for j ≥ j0, and hence |p(z0)| 1 0 < ≤|p(zj )| < for all j ≥ j0. 2 nkj →∞ →∞ But this is impossible, since nkj as j . Hence there exists δ>0 such that p ueit ≥ δ for all t ∈ [−π, π]andu ∈ [−α, α]. It follows that dαd−1 + ···+ |c | |D h(t, u)|≤ 1 for all t ∈ [−π, π]andu ∈ [−α, α]. 2 δ2 Thus the hypothesis of Proposition B.1 is satisfied with E =[−α, α]. Since α>0 is' arbitrary, we see that the integral function H : R → C given by π H(u)= − h(t, u)dt is differentiable and π & π H (u)= D2h(t, u)dt for all u ∈ R. −π − it 2 ∈ − If u =0,then D 2'h(t, u)=D2 h(t, 0) = c1e /c0 for t [ π, π], and so − 2 π it − 2 iπ − −iπ H (0) = c1/c0 − e dt = c1/ic0 e e =0,whereas π d idt it − u ide + ···+ c1uie 1 D2h(t, u)= = D1h(t, u), if u =0 . d idt it 2 iu (u e + ···+ c1ue + c0) iu Consequently, for all u ∈ R with u =0, & π 1 1 H (u)= D1h(t, u)dt = [h(π, u) − h(−π, u)] = 0, iu −π iu where the second equality follows from the Fundamental Theorem of Calculus (part (ii) of Proposition 6.24) and the last equality follows since eπi = e−πi. Thus H(u) = 0 for all u ∈ R. Hence by Corollary 4.23, H is a constant function, and so & π 2π H(u)=H(0) = h(t, 0)dt = = 0 for all u ∈ R. −π c0 On the other hand, we show that H(n) → 0asn →∞. To this end, let 1 g (t):=h(t, n)= for n ∈ N and t ∈ [−π, π]. n p(neit) Now for all n ∈ N and t ∈ [−π, π], | it | | d idt ··· it | p(ne ) = n e + + c1ne + c0 d idt d−1 i(d−1)t it ≥ n e − c − n e + ···+ c ne + c d 1 1 0 d − cd−1 i(d−1)t ··· c1 it c0 = n 1 e + + − e + n nd 1 nd |c − | |c | |c | ≥ nd 1 − d 1 −···− 1 − 0 n nd−1 nd nd ≥ , provided n ≥ 2(|c | + |c | + ···+ |c − |) . 2 0 1 d 1 522 B Fundamental Theorem of Algebra This implies that gn(t) → 0asn →∞for each t ∈ [−π, π], and also that the sequence (|gn|) is uniformly bounded on [−π, π]. So by applying the Arzel`a Bounded Convergence Theorem (Proposition 10.40) to the real and imaginary parts of gn,weobtain & & & π π π H(n)= h(t, n)dt = gn(t)dt −→ 0 dt =0. −π −π −π But since H(u)=2π/c0 = 0, this is a contradiction. It follows that p(z)must have a root in C. Corollary B.4. (i) Every nonzero polynomial p(z) of degree d with coeffi- cients in C can be factored into d linear factors as p(z)=c(z − α1) ···(z − αd), where c ∈ C with c =0 and α1,...,αd are complex numbers. (ii) Every nonzero polynomial p(z) of degree d with coefficients in C isaprod- uct of powers of distinct linear factors as m1 mk p(z)=c(z − λ1) ···(z − λk) , where c ∈ C with c =0 and λ1,...,λk are distinct complex numbers and m1,...,mk are positive integers satisfying m1 + ···+ mk = d. (iii) (Real Fundamental Theorem of Algebra) Every nonzero polynomial p(x) with coefficients in R can be factored as a finite product of linear polynomials and quadratic polynomials with negative discriminants. Proof. (i) The result is obvious when d = 0, since an empty product equals 1. For d ≥ 1, it follows from Proposition B.3 by induction on d. (ii) This follows from (i) by collating equal linear factors. 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List of Symbols and Abbreviations Definition/Description Page N set of all positive integers 1 Z set of all integers 1 Q set of all rational numbers 1 R set of all real numbers 2 sum 3 product 3 A := BAis defined to be equal to B 3 R+ set of all positive real numbers 3 sup S supremum of a subset S of R 4 inf S infimum of a subset S of R 4 max S maximum of a subset S of R 5 min S minimum of a subset S of R 5 [x] integer part of a real number x 6 x integer part or the floor of a real number x 6 √x ceiling of a real number x 6 √n anth root of a nonnegative real number a 7 a square root of a nonnegative real number a 7 m | nmdivides n 8 m nmdoes not divide n 8 (a, b) open interval {x ∈ R : a © Springer Nature Switzerland AG 2018 527 S. R. Ghorpade, B. V. Limaye, A Course in Calculus and Real Analysis, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-030-01400-1 528 List of Symbols and Abbreviations Definition/Description Page A.M. arithmetic mean 12 G.M. geometric mean 12 D \ C {x ∈ D : x ∈ C} 13 D × E {(x, y):x ∈ D and y ∈ E} 14 f|C restriction of f : D → E toasubsetC of D 16 g ◦ f composite of g with f 16 f −1 inverse of an injective function f 16 f =0 f is the zero function (on its domain) 17 f ≤ gf(x) ≤ g(x) for all x 17 f ≥ 0 f(x) ≥ 0 for all x 17 R[x] set of all polynomials in x with coefficients in R 18 deg p(x) degree of a nonzero polynomial p(x)18 IVP intermediate value property 29 H.M. harmonic mean 34 GCD greatest common divisor 37 LCM least common multiple 37 (an) sequence whose nth term is the real number an 41 an → a sequence (an) tends to a real number a 42 lim an limit of the sequence (an)43 n→∞ an = O(bn)(an)isbig-ohof(bn)51 an = o(bn)(an) is little-oh of (bn)52 an ∼ bn (an) is asymptotically equivalent to (bn)52 an →∞ sequence (an) tends to ∞ 52 an →−∞ sequence (an) tends to −∞ 52 → does not tend to 53 lim sup an limit superior of (an)60 n→∞ lim inf an limit inferior of (an)60 n→∞ lim f(x) limit of f(x)asx tends to c 84 x→c lim f(x) left limit of f(x)asx tends to c 89 x→c− lim f(x) right limit of f(x)asx tends to c 90 x→c+ f(x)=O(g(x)) f(x) is big-oh of g(x)asx →∞ 92 f(x)=o(g(x)) f(x) is little-oh of g(x)asx →∞ 92 f(x) ∼ g(x) f(x) is asymptotically equivalent to g(x)93 df f (c), derivative of f at c 106 dx x =c 2 d f f (c), second derivative of f at c 114 dx2 x=c n (n) d f f (c), n nth derivative of f at c 114 dx x=c List of Symbols and Abbreviations 529 Definition/Description Page f−(c) left derivative of f at c 115 f+(c) right derivative of f at c 115 MVT Mean Value Theorem 122 ≈ approximately equal 126 L’HR L’Hˆopital’s Rule 135, 136 Pn partition of [a, b]inton equal parts 182 m(f) infimum of {f(x):x ∈ [a, b]} 182 M(f) supremum of {f(x):x ∈ [a, b]} 182 mi(f) infimum of {f(x):x ∈ [xi−1,xi]} 182 Mi(f) supremum of {f(x):x ∈ [xi−1,xi]} 182 L(P, f)lowersumforf with respect to P 183 U(P, f) upper sum for f with respect to P 183 L(f) lower Riemann integral of f 183 ' U(f) upper Riemann integral of f 183 b a f(x)dx Riemann integral of f on [a, b] 185 f + positive part of f 202 f − negative part of f 202 ' FTC Fundamental Theorem of Calculus 204 " f#(x)dx an indefinite integral of f 206 b b − F (x) a, F (x) a F (b) F (a) 206 S(P, T ,f) Riemann sum for f corresponding to a partition P and a tag set T 211 μ(P ) mesh of a partition P 211 (D) length of a bounded subset D of R 221 ln logarithmic function 234 e unique real number such that ln e = 1 234 exp exponential function 236 arctan arctangent function 246 π 2sup{arctan x : x ∈ (0, ∞)} 247 ∠(OP1,OP2) angle between OP1 and OP2 269 L1 L2 lines L1 and L2 are parallel 271 L1 ∦ L2 lines L1 and L2 are not parallel 271 (L1,L2) (acute) angle between L1 and L2 271 L1 ⊥ L2 lines L1 and L2 are perpendicular 272 L1 ⊥ L2 lines L1 and L2 are not perpendicular 272 (C1,C2; P )angleatP between C1 and C2 273 Area (R) area of a region R 296 Vol (D) volume of a solid body D 303, 307 (C) length of a curve C 316 Area (S) area of a surface S 325 Av(f) average of f 329 Av(f; w) weighted average of f with respect to w 329 530 List of Symbols and Abbreviations Definition/Description Page (x, y) centroid of a curve or a planar region 330, 333 (x, y,z) centroid of a surface or a solid body 331, 334 Q(f) Quadrature Rule for f 341 R(f) Rectangular Rule for f 341 M(f) Midpoint Rule for f 341 T (f) Trapezoidal Rule for f 342 S(f) Simpson Rule for f 342 Rn(f) Compound Rectangular Rule for f 343 Mn(f) Compound Midpoint Rule for f 343 Tn(f) Compound Trapezoidal Rule for f 344 Sn(f) Compound Simpson Rule for f 345 k≥1 ak series whose sequence of terms is (ak) 366 ∞ ak sum of ≥ ak, when convergent 366 ' k=1 k 1 ∞ t≥a f(t)dt improper integral of f on [a, ) 391 ' ∞ ' f(t)dt value of f(t)dt, when convergent 392 ' a t≥a −∞ t≥b f(t)dt improper integral of f on ( ,b] 406 ' ' b f(t)dt value of f(t)dt, when convergent 406 '−∞ t≥b −∞ ∞ R f(t)dt improper integral of f on ( , ) 406 ' ∞ ' f(t)dt value of f(t)dt, when convergent 406 ' −∞ R a Definition/Description Page Bn(f) nth Bernstein polynomial function associated with f 449 ak(f),bk(f) Fourier coefficients of f 453 Sn(f) nth partial sum of the Fourier series of f 453 σn(f) arithmetic mean of S0(f),S1(f),...,Sn(f) 453 Dn nth Dirichlet kernel 454 Kn nth Fej´er kernel 454 fn → f boundedly (fn) converges boundedly to f 458 f(·,u) function given by t −→ f(t, u) for a fixed u 467 · −→ ' f(t, ) function given by u f(t, u) for' a fixed t 467 b · −→ b a f(t, )dt integral function given by u a f(t, u)dt 467 D1f(·,u) derivative of the function f(·,u) for a fixed u 469 ' D2f(t, ·) derivative of the function f(t, ·) for a fixed t 469 · ∞ ≥ f(t, )dt improper integral of f on [a, ) depending on a parameter 472 't a ' ∞ · −→ ∞ a f(t, )dt improper integral function given by u a f(t, u)dt 472 Fs(f) Fourier sine integral of f 476 Fc(f) Fourier cosine integral of f 476 ' L(f) Laplace integral of f 477 · ≤ f(t, )dt improper integral of f on (a, b] depending on a parameter 486 a α-δ condition, 93 archimedian ordered field, 513 β-δ condition, 94 archimedian property, 5, 509 -α condition, 91 arctangent function, 246, 247 -δ condition, 72, 83, 88 area, 185, 296 γ, 279 arithmetic–geometric mean, 63 π, 247 Arzel`a Theorem, 461, 462 e, 234 asymptote, 94 kth Term Test, 372 asymptotic error constant, 180 nth term, 505 asymptotically equivalent, 93 attains bounds, 23 A.M.-G.M. inequality, 12 average, 328, 329 A.M.-H.M. inequality, 38 Abel kth Term Test, 372, 417 base, 239 Abel Inequality, 34 base of the natural logarithm, 243 Abel Lemma, 382 basic inequalities for absolute values, 10 Abel Test, 419, 421, 494, 496 basic inequalities for powers and roots, Abel Theorem, 498, 499 10 abscissa of convergence, 443, 478 basic inequality for rational powers, 38 absolute extremum, 150 basic inequality for Riemann integrals, absolute maximum, 149 186 absolute minimum, 149 Bernstein polynomial function, 449 absolute value, 10, 513, 517 Bernstein Theorem, 450 absolutely convergent, 370, 396 beta function, 413, 502 accumulation point, 83 bijective, 15 acute angle, 269 binomial coefficient, 32 algebraic function, 20 Binomial Inequality, 11, 38 algebraic number, 22 binomial series, 390, 423 alternating series, 368 Binomial Theorem, 6, 33, 449 angle, 269 Bliss Theorem, 231 angle between two curves, 273 Bolzano–Weierstrass Theorem, 55 angle between two lines, 271 boundary, 220 antiderivative, 205 boundary point, 150 arc length, 316, 318 bounded, 4, 22, 505, 518 © Springer Nature Switzerland AG 2018 533 S. R. Ghorpade, B. V. Limaye, A Course in Calculus and Real Analysis, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-030-01400-1 534 Index bounded above, 4, 22, 505 concave upward, 24 bounded below, 4, 22, 505 conditionally convergent, 371, 396 bounded convergence, 458, 464, 486 congruence modulo n, 505 bounded interval, 9 conjugate, 517 bounded variation, 417 constant function, 16 constant polynomial, 18 Carath´eodory Lemma, 108 content zero, 218 cardioid, 267 continuous, 67, 518 Cartesian coordinates, 266 Continuous Inverse Theorem, 79 Cartesian equation, 267 continuously differentiable, 120 Cauchy completeness, 58, 516 contraction, 179 Cauchy Condensation Test, 417 contractive, 179 Cauchy condition, 232 Convergence of Newton sequences, 170 Cauchy Criterion, 58, 89, 91, 368, 394, Convergence Test for Fourier Integrals, 430, 439, 474 405 Cauchy form of remainder, 148 Convergence Test for Trigonometric Cauchy Mean Value Theorem, 134 Series, 380 Cauchy principal value, 406, 408 convergent, 42, 366, 392, 406–408, 411 Cauchy product, 422 converges, 42, 366, 392 Cauchy Root Test, 376 converges boundedly, 458, 464, 485 Cauchy sequence, 56, 505 converges pointwise, 426, 438, 472 Cauchy–Schwarz Inequality, 12 converges uniformly, 429, 438, 473 ceiling, 6 convex, 24 ceiling function, 21 coprime, 8 centroid, 330, 331, 333 cosecant function, 255 Ces`aro convergent, 53, 371 cosine function, 250 Ces`aro mean, 371 cotangent function, 256 Ces`aro sum, 371 countable, 15 Ces`aro summable, 371 countably infinite, 15 Chain Rule, 113 criss-cross function, 80 closed interval, 9 critical point, 150 closed set, 73 cubic polynomial, 18 cluster point, 59 cut, 515 codomain, 14 coefficient, 18, 382 D’Alembert Ratio Test, 375 common refinement, 183 decimal expansion, 65 Comparison Test, 373, 401, 410, 464 decreasing, 24, 47 complete ordered field, 513 Dedekind cut, 515 completeness property, 5, 510 definite integral, 206 complex exponential, 518 degree, 18, 20 complex number, 517 degree measure, 270 complex-valued function, 518 denumerable, 15 composite, 16 derivative, 106, 518 Compound Midpoint Rule, 343 differentiable, 106, 518 Compound Rectangular Rule, 343 Differentiable Inverse Theorem, 114 Compound Simpson Rule, 345 digamma function, 502 Compound Trapezoidal Rule, 344 digit, 65 concave, 24 Dini Theorem, 432 concave downward, 24 Dirichlet function, 68, 186 Index 535 Dirichlet kernel, 454 Fourier sine integral, 476 Dirichlet series, 442 FTC, 202 Dirichlet Test, 379, 404, 422, 440, 464, function, 14 475, 500 Fundamental Theorem of Calculus, 204 discontinuous, 67 Fundamental Theorem of Riemann discriminant, 19 Integration, 207 disk method, 310 divergent, 42, 366, 392, 406–408 G.M.-H.M. Inequality, 34 diverges, 52, 366, 392 gamma function, 414, 487 domain, 14 GCD, 37, 39 domain additivity, 190, 229 Generalized AM-GM Inequality, 286 Dominated Convergence Theorem, 435 Generalized Binomial Inequality, 11 geodesic, 320 elementary function, 277 geometric series, 367 elementary transcendental function, 274 graph, 13 endpoint, 9 great circle, 321 enumeration, 15 greatest common divisor, 37, 39 equivalence class, 504 greatest lower bound, 4 equivalence relation, 504 grouping of terms, 423 error function, 363 growth rate, 52, 93 error in linear approximation, 160 error in quadratic approximation, 161 H¨older Inequality, 286 Euler constant, 279 half-open interval, 9 Euler Summation Formula, 398 harmonic mean, 34 even function, 16 harmonic series, 367 exponent, 239 Hausdorff property, 34 exponential function, 236 helix, 319 exponential series, 367 horizontal asymptote, 94 Extended Mean Value Inequality, 125 hyperbolic cosine, 280 Extended Mean Value Theorem, 124 hyperbolic sine, 280 extended real numbers, 9 hypergeometric series, 418 factorial, 32 identity function, 16 Fej´er kernel, 454 imaginary part, 517, 518 Fej´er Theorem, 456 implicit differentiation, 117 Fichtenholz–Lichtenstein Theorem, 468 implicitly defined curve, 116 field, 512 improper integral, 391, 407 finite set, 15 improper integral function, 472 First Derivative Test, 153, 154 increasing, 24, 47 fixed point, 163 increment function, 109 floor, 6 indefinite integral, 206 floor function, 21 induction, 33 folium of Descartes, 358 inequality for rational roots, 38 for all large k ∈ N, 366 infimum, 4 for all large t ∈ [a, ∞), 391 infinite interval, 9 Fourier coefficient, 453 infinite series, 365 Fourier cosine integral, 476 infinite set, 15 Fourier integral, 476 infinitely differentiable, 114 Fourier series, 453 infinity, 9 536 Index injective, 15 linear polynomial, 18 instantaneous speed, 106 Lipschitz condition, 203 instantaneous velocity, 106 local extremum, 26, 153 integer part, 6, 513 local maximum, 26 integer part function, 21 local minimum, 26 integrable, 185, 216, 518 log-convex, 415 integral, 518 log-convexity of the gamma function, integral function, 467 424 Integral Test, 397 logarithmic function, 234 Integration by Parts, 208 logarithmic function with base a, 243 Integration by Substitution, 209 lower bound, 4 interior point, 106 lower Riemann integral, 185 Intermediate Value Property, 29 lower sum, 183 Intermediate Value Theorem, 79 interval, 9 Maclaurin series, 386 interval of convergence, 383 map, 14 inverse function, 16 maximum, 5 inverse trigonometric function, 256 Mean Value Inequality, 122 irrational number, 7 Mean Value Theorem, 122 IVP, 29, 120 Mean Value Theorem for Integrals, 231, 232 Jensen Inequality, 36 mesh, 211 Midpoint Rule, 341 L’Hˆopital’sRule, 135, 136 minimum, 5 Lagrange form of remainder, 125 Minkowski Inequality, 286 Lagrange Identity, 13 modulus, 10, 517 Laplace integral, 477 monic polynomial, 18, 519 laws of exponents, 6, 245 monotone completeness, 516 laws of indices, 6, 245 Monotone Convergence Theorem, 435, LCM, 37 460 leading coefficient, 18 monotonic, 24, 47 least common multiple, 37 monotonic sequence of functions, 432 least upper bound, 4 monotonically decreasing, 24, 47 left (hand) derivative, 115 monotonically increasing, 24, 47 left (hand) limit, 89 multiplicity, 146, 147 Leibniz Rule, 115, 228 MVT, 119 Leibniz Test, 380, 441, 465 lemniscate, 268 natural logarithm, 234 length, 218, 221, 318, 363 natural number, 1 lima¸con, 268 negative, 3 limit, 42, 84, 91 negative part, 202 Limit Comparison Test, 373, 402 Nested Interval Theorem, 66 limit inferior, 60 Newton method, 169 limit of composition, 101 Newton sequence, 169 limit point, 83 Newton–Raphson method, 169 limit superior, 60 nodes, 341 Limit Theorem, 43, 86 nonconstant polynomial, 519 linear approximation, 159 nonexpansive, 179 linear convergence, 179 nonnegative function, 17 Index 537 normal, 117, 118 power, 6, 7, 239 null sequence, 506 power function, 241, 243 number line, 1 power mean, 39 power mean inequality, 291 oblique asymptote, 95 power series, 382, 446 obtuse angle, 269 prime number, 37 oddfunction,16 primitive, 205 one-dimensional content zero, 218 Prismoidal Formula, 363 one-one, 15 one-to-one correspondence, 15 quadratic approximation, 161 onto, 14 quadratic convergence, 180 open interval, 9 quadratic polynomial, 18 order completeness, 516 quadrature rule, 341 order isomorphism, 513 quotient, 17 order of convergence, 180 quotient rule, 111 ordered field, 512 orthogonal intersection of curves, 273 Raabe Test, 377, 418 orthogonality relations, 282 radian measure, 270 radius of convergence, 383 P´olya Theorem, 497 range, 14 Pappus Theorem, 337, 338 Ratio Test, 375 parameter domain, 307 rational function, 19, 20 parametrically defined curve, 116 rational number, 1 parity, 504 real analytic function, 391 partial fraction decomposition, 19 Real Fundamental Theorem of Algebra, partial integral, 392 19, 522 partial integral function, 392, 472 real part, 517, 518 partial sum function, 438 real-valued function, 17 Partial Summation Formula, 379 rearrangement of terms, 423 partition, 182 reciprocal, 3 Pascal triangle, 32 rectangular coordinates, 266 Pascal triangle identity, 32 Rectangular Rule, 341 periodic, 227 rectifiable, 363 perpendicular lines, 272 recurring decimal expansion, 65 Picard Convergence Theorem, 165 reduced form, 8 Picard method, 165 refinement, 183 Picard sequence, 165 reflexive, 504 piecewise smooth, 318 relation, 504 point of inflection, 26, 157 relatively prime, 8 pointwise bounded, 492 remainder, 125 pointwise convergence, 426, 438, 472 residue classes modulo n, 505 pointwise equicontinuous, 499 restriction, 16 pointwise limit, 426 rhodonea curves, 268 polar coordinates, 266 Riemann condition, 188 polar equation, 267 Riemann integral, 185, 216 polynomial, 17 Riemann sum, 211 polynomial function, 20 right (hand) derivative, 115 positive, 508 right (hand) limit, 90 positive part, 202 right angle, 269 538 Index Rolle Theorem, 121 tangent function, 249 root, 7, 18, 147 tangent line approximation, 160 root mean square, 39 Taylor Formula, 125 Root Test, 376, 402 Taylor polynomial, 125 rose, 268 Taylor series, 386 Taylor Theorem, 124, 230 Sandwich Theorem, 45, 87, 368, 394 telescoping series, 369 Schl¨omilch form of remainder, 148 tends, 52 secant function, 255 term, 41 second derivative, 114 Test for Uniform Convergence, 429, 439, Second Derivative Test, 154 473 semiopen interval, 9 Thomae function, 88, 146, 195 sequence, 41, 490, 505 transcendental function, 21, 274 sequence of partial sums, 366 transcendental number, 22 sequence of terms, 366 transitive, 504 series, 365, 490 Trapezoidal Rule, 342 shell method, 311 Triangle Inequality, 10, 518 signed angle, 270 trigonometric function, 256 Simpson Rule, 342 trigonometric polynomial function, 452 sine function, 250 trigonometric series, 441, 452 slice, 302 sliver, 306 unbounded interval, 9 smooth, 316 unbounded set, 4 solid angle, 326 uncountable, 15 solid of revolution, 309 uniform Ces`aro convergence, 456, 498 spiral, 267 uniform convergence, 429, 438, 442, 446, steradian, 327 473, 476, 477 Stirling Formula, 288 uniform limit, 429 strict local extremum, 27 uniformly bounded, 430 strict local maximum, 27 uniformly Cauchy, 430 strict local minimum, 27 uniformly continuous, 81 strict point of inflection, 27 uniformly equicontinuous, 499 strictly concave, 26 upper bound, 4 strictly convex, 26 upper Riemann integral, 185 strictly decreasing, 26 upper sum, 183 strictly increasing, 26 strictly monotonic, 26 valueofanimproperintegral,392, subinterval, 9 406–408 subinterval induced by a partition, 182 vertical asymptote, 95 subsequence, 54 Wallis Formula, 287 sum, 366, 438 washer method, 310 sum function, 438 Weierstrass Approximation Theorem, supremum, 4 451, 457 surface area, 307 Weierstrass M-Test, 439, 474, 486 surjective, 14 weight, 341 symmetric, 16, 504 weight function, 329 tag set, 211 weighted average, 329 tail of a series, 366 zero function, 17 tail of an improper integral, 392 zero polynomial, 18 tangent, 117