Pringsheim Convergence and the Dirichlet Function

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, { } : , mn a n m∈ may be defined so that the iterated limits lim lim m n mn a →∞ →∞ and lim lim n m mn a →∞ →∞ exist and are equal for all x, and yet the Pringsheim limit ( ) ( ) , , lim mn m n a → ∞ ∞ does not exist. The sequence ( ) { } 2 cos ! n m x π is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere 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.


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 behavior of the iterated limits as well as the limits of ordinary sequences induced by collapsing the two indices into one according to a suitable functional dependence (e.g.re-index 2 mn ).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-tions of the form ( ) ( ) in this context.One iterated limit of this sequence, namely ( ) , is a well-known example of the construction of the Dirichlet "salt-and-pepper" function ( ) . In addition to establishing a theorem on Pringsheim convergence which is useful in its own right, we will be able to conclude that ( ) { } mn f x does not converge pointwise in this sense.
Moreover, it will be shown that there are irrational numbers for which the ordinary sequence ( ) { } 2 cos π !m m x does not converge to zero.

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 { }
: , mn a m n∈  , we say it converges to the limit L if for every preassigned 0 ε > there exists a ( ) . This situation will be denoted by In this definition, it is understood that m and n are to exceed ??
In this array the rows represent fixed m with n increasing, and the columns represent fixed n with m increasing.= may or may not be equal.A trivial but illustrative case is given by the double sequence . The array with iterated limits is: 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 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: This array shows that a double sequence can be Pringsheim convergent, and although none of the row 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.

Main Theorem
. This defines a ( ) , where : φ →   is strictly monotone increasing.The terminology is suggested by the fact that for any ( ) will eventually enter and stay inside the part of the square below (south) and to the right (east) of a KK .The ordinary subsequences where either m or n are held constant (the horizontal or vertical subsequences in the array) do not have this property.Clearly, every southeastern subsequence can be converted to an ordinary subsequence.

Corollary 1a: If { }
: , mn a n m∈  is a double sequence of real numbers, then

and only if every southeastern subsequence of a mn converges to L.
Proof: (Necessity) Suppose every southeastern subsequence of a mn converges to L. Fix 0 ε > and assume for the sake of contradiction that , we may choose j j ′ > such that ( ) ( ) > by the isotonicity of φ .In any case, by relabelling, if necessary, we may arrange that ( ) x n m ∈  converges pointwise in the Pringsheim sense if whenever x is fixed, the resulting numerical sequence converges in the regular Pringsheim sense (definition 1).This situation will be denoted by . If a doubly indexed sequence of functions were simply pointwise convergent and the iterated limits did not commute, the limit function would be ill-defined.Our definition along with Theorem 1 resolves this issue.
Corollary 1b: Theorem 1 applies to the pointwise limits of doubly indexed sequences of functions with a mn replaced by , we have ( ) x δ is everywhere discontinuous, and a fanciful image of its "graph" has given rise to the name "salt-and-pepper" function.It is often used as an example of a function that is Lebesgue integrable but not Riemann integrable (although integrable in the generalized Riemann sense).
Dirichlet's function turns out to be an example of a Baire class 2 function.Recall that Baire class 0 consists of functions that are continuous.Baire class 1 functions are pointwise limits of sequences of Baire class 0 functions.
In general, a Baire class α function is the pointwise limit of a sequence of functions from the union of all Baire classes with indices less than α , where the class index is allowed to range over the countable ordinals.
The fact that the Dirichlet function cannot be expressed as the limit of a sequence of continuous functions will play a key role in establishing our claim that does not converge in the Pringsheim sense.

Baire's Category Theorem (BCT)
René-Louis Baire proved the seminal theorem that bears his name in 1899 as part of his doctoral dissertation [4].He introduced the famously bland terminology Category 1 for meager sets and Category 2 for non-meager sets.
The column to the extreme right records lim n these limits exist.For arbitrary 0 ε > 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 .
= .Note also that if we violate the condition that m and n exceed ( ) K ε independently by setting m n = , lim m mm a →∞  be a double sequence of real numbers with Pringsheim limit If for some , M N ∈  both the partial limit lim n and are equal to L. Proof: Without restriction of generality, consider the column sequence formed by the partial limits A m for m > M. Fix ε > 0. We claim that lim m  converges to L in the sense of Pring- sheim, there exists( )

ε
Pringsheim square for which all of the row and column partial lim- its exist and every a mn within the square differs absolutely from L by less than 2 In view of the array we have used to visualize the Pringsheim definition, let us call a double subsequence

2 a-
Construct a southeastern subsequence of "bad" terms as follows: Let K = 1. in the (1)-Pringsheim square (the entire array) such that 1 square so that 2 a L ε − ≥ .Likewise a 3 from the ( ) Pringsheim square and so forth recursively.By the manner of construction, i > j requires m > m j and n i > n j , so the resulting subsequence is certainly southeastern.Moreover, { } i a cannot converge to L. The contradiction establishes necessity.(Sufficiency) Suppose { } : , mn a n m∈  is a double sequence of real numbers with Fix 0 ε > and consider the southeastern subsequence { } jk a .By Pringsheim convergence, there exists a ( )

2 :
Pringsheimsquare that constrains the absolute difference between jk a and L to be less than ε .Again by the isotonicity of φ , − < for j j * ≥ .Hence the southeastern sequence { } jk a converges to L.  Let us formulate a definition of pointwise Pringsheim convergence of functions so that we have a basis for studying m ∈  .In particular, we will consider The doubly indexed sequence of real functions x ∈  .If x ∈  , and !m x is not an integer, then However, once m is sufficiently large, !m x becomes and remains an integer, hence cos π ! 1 m x = ± for those cases.It follows that x.On the other hand, if x ∉  , the quantity cos π !m x is never an integer for any m, hence