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Pringsheim Convergence and the Dirichlet Function

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Pringsheim Convergence and the Dirichlet Function Advances in Pure Mathematics, 2016, 6, 441-445 Published Online May 2016 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2016.66031 Pringsheim Convergence and the Dirichlet Function Thomas Beatty, Bradley Hansen Department of Mathematics, Florida Gulf Coast University, Ft. Myers, FL, USA Received 3 April 2016; accepted 21 May 2016; published 24 May 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Double sequences have some unexpected properties which derive from the possibility of com- muting limit operations. For example, {amn :, nm∈ } may be defined so that the iterated limits limm→∞ lim n →∞a mn and limn→∞ lim m →∞a mn exist and are equal for all x, and yet the Pringsheim limit π 2n lim(mn,,)→∞∞( )amn does not exist. The sequence {(cosmx ! ) } is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an every- where discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct. Keywords Convergence, Pointwise Limit, Double Sequence, Pringsheim, Dirichlet Function, Baire Category Theorem, Cosine 1. Introduction The problem of convergence of a doubly indexed sequence presents some interesting phenomena related to the order of taking iterated limits as well as subsequences where one index is a function of the other. Convergence of a double sequence in the sense of Pringsheim is a strong enough condition to allow us to characterize the be- havior of the iterated limits as well as the limits of ordinary sequences induced by collapsing the two indices into −mn 2 one according to a suitable functional dependence (e.g. re-index amn = 2 by setting nm= to obtain −m3 am = 2 ). We will show that an unconditional converse establishing convergence in the Pringsheim sense from properties of the iterated limits is not obtainable. We can easily extend the notion of Pringsheim convergence of numerical sequences to pointwise convergence in the Pringsheim sense for functions. Our main goal is to investigate the doubly indexed sequence of real func- How to cite this paper: Beatty, T. and Hansen, B. (2016) Pringsheim Convergence and the Dirichlet Function. Advances in Pure Mathematics, 6, 441-445. http://dx.doi.org/10.4236/apm.2016.66031 T. Beatty, B. Hansen 2n tions of the form fmn ( x) = (cos π mx! ) in this context. One iterated limit of this sequence, namely limm→∞ lim n →∞ fx mn ( ) , is a well-known example of the construction of the Dirichlet “salt-and-pepper” function 1 if x ∈ δ ( x) . Recall δ ( x) = . In addition to establishing a theorem on Pringsheim convergence which is 0 if x ∉ useful in its own right, we will be able to conclude that { fxmn ( )} does not converge pointwise in this sense. 2m Moreover, it will be shown that there are irrational numbers for which the ordinary sequence {(cos πmx! ) } does not converge to zero. 2. Background The German mathematician Alfred Pringsheim formulated the following definition of convergence for double sequences in 1897 [1]. Definition 1: Given the doubly indexed sequence {amn :, mn∈ } , we say it converges to the limit L if for every preassigned ε > 0 there exists a K (ε )∈ such that aLmn −<ε whenever mn, > K(ε ) . This situ- ation will be denoted by lim(mn,,)→∞∞( ) aLmn = . In this definition, it is understood that m and n are to exceed K (ε ) independently. Specifically, there should be no functional relationship between m and n, such as mn= 2 , for example. The definition lends itself to an intuitively appealing visual. We will call the semi-infinite set of grid points {amn :, mn> K(ε )} below and to the right of aKK a K (ε ) Pringsheim square: aaaa11 12 13 14 a1Kn a1 → A1 → aaaa21 22 23 24 a2Kn a2 A2 aaaa a a→ A 31 32 33 34 3Kn3 3 aaaa41 42 43 44 a4Kn a4 → A4 →∗ aaaaKKKK1234 aKK → AK →∗ aaaam1234 m m m amn → Am ↓ ↓ ↓ ↓↓↓↓↓ ↓ BBBB1234∗ BKn ∗→ B ?? In this array the rows represent fixed m with n increasing, and the columns represent fixed n with m increasing. The column to the extreme right records limn→∞ aA mn= m and the row at the bottom records limm→∞ aB mn= n , whenever these limits exist. For arbitrary ε > 0 the double limit L exists if there is a K (ε ) such that the ab- solute difference between L and any term in the K (ε ) Pringsheim square is strictly less than ε . The iterated limits limm→∞ lim n →∞ aA mn= lim m→∞ m and limn→∞ lim m →∞ aB mn= lim n→∞ n may or may not be equal. A trivial 1 if mn= but illustrative case is given by the double sequence amn = . The array with iterated limits is: 0 if mn≠ 1000 0 0→ 0 0100000→ 0010000→ 0001 0 0→ 0 00 → 0 000001 0→ 0 → 0 000000 1→ 0 ↓↓↓↓↓↓↓↓ ↓ 00000000→ ?! 442 T. Beatty, B. Hansen Observing that the iterated limits exist and are equal to zero, but the double limit in the Pringsheim sense does not even exist, since for any K (ε ) there are terms with mm,′′ ,, nn> K(ε ) such that amn = 1 and amn′′= 0 . Note also that if we violate the condition that m and n exceed K (ε ) independently by setting mn= , limm→∞ a mm exists and equals 1. This example immediately dashes any hope of establishing a Fubini-like result where if the two iterated limits exist and are equal then the double limit in the Pringsheim sense exists and is the same. A more optimistic case is this: 111 1 −−10 − − − → 234 n 1 1 1 11 1 − 0 −→ 2 64 2n 2 1 1 1 5 21 2 − −→ 3 6 3 12 3 n 3 1 1 5 1 31 3 − −→ 4 4 12 2 4 n 4 → 1112131 mn+ −−−− 11 − → mmmm234 mn ↓↓↓↓↓↓ ↓ 123 1 0 11−→ 234 n This array shows that a double sequence can be Pringsheim convergent, and although none of the row (limn→∞ a mm ) or column (limm→∞ a mm ) partial limits equal the Pringsheim limit (1), the respective row-first (limm→∞ lim n →∞ a mn ) and column-first (limn→∞ lim m →∞ a mn ) iterated limits can equal the Pringsheim limit. Motivated by this example we formulate a theorem that connects Pringsheim convergence to the existence and equality of the associated iterated limits. 3. Main Theorem Theorem 1: Let {amn :, nm∈ } be a double sequence of real numbers with Pringsheim limit lim(mn,,)→∞∞( ) aLmn = . If for some MN, ∈ both the partial limit limn→∞ aA mn= m exists for mM> and the partial limit limm→∞ aB mn= n exists for nN> , then the iterated limits limm→∞ lim n →∞ a mn and limn→∞ lim m →∞ a mn exist and are equal to L. Proof: Without restriction of generality, consider the column sequence formed by the partial limits Am for m > M. Fix ε > 0. We claim that limmm→∞ AL= . Since {amn :, nm∈ } converges to L in the sense of Pring- ε sheim, there exists K (ε ) such that mn, > K(ε ) implies aL−<. Increase K (ε ) , if warranted, so that mn 2 K(ε ) ≥ max[ MN , ] . This defines a K (ε ) Pringsheim square for which all of the row and column partial lim- ε its exist and every amn within the square differs absolutely from L by less than . Now 2 ε ALAa−≤ − + a − L, and since lim →∞ aA= , we may stipulate that Aa−<. It is clear that m m mn mn n mn m m mn 2 εε AL−<+=ε , and our claim that lim→∞ A= lim→∞ lim →∞ aL= is established. The same argu- m 22 m m m n mn ment mutatis mutandi shows that limn→∞ B n= lim n→∞ lim m →∞ aL mn = . In view of the array we have used to visualize the Pringsheim definition, let us call a double subsequence {aajk} ⊂ { mn} southeastern if kj= φ ( ) , where φ : → is strictly monotone increasing. The terminology is ε suggested by the fact that for any K ( ) Pringsheim square the double subsequence {a jjφ( )} will eventually enter and stay inside the part of the square below (south) and to the right (east) of aKK. The ordinary subse- quences where either m or n are held constant (the horizontal or vertical subsequences in the array) do not have 443 T. Beatty, B. Hansen this property. Clearly, every southeastern subsequence can be converted to an ordinary subsequence. Corollary 1a: If {amn :, nm∈ } is a double sequence of real numbers, then lim(mn,,)→∞∞( ) aLmn = if and only if every southeastern subsequence of amn converges to L. Proof: (Necessity) Suppose every southeastern subsequence of amn converges to L. Fix ε > 0 and assume for the sake of contradiction that lim(mn,,)→∞∞( ) aLmn ≠ . Then every K (ε ) Pringsheim square contains an amn such that aLmn −≥ε . Construct a southeastern subsequence of “bad” terms as follows: Let K = 1. Choose a term aa= in the (1)-Pringsheim square (the entire array) such that aL−≥ε . Now select aa= in 1 mn11 1 2 mn22 the (a1 ) -Pringsheim square so that aL2 −≥ε . Likewise select a3 from the (a2 ) -Pringsheim square and so forth recursively. By the manner of construction, i > j requires mi > mj and ni > nj, so the resulting subsequence is certainly southeastern. Moreover, {ai} cannot converge to L. The contradiction establishes necessity. (Sufficiency) Suppose {amn :, nm∈ } is a double sequence of real numbers with lim(mn,,)→∞∞( ) aLmn = .
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