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We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.

Let X be a nonempty set,

Obviously, if

doesn’t hold in general for an arbitrary (infinite) set X; i.e., the pointwise convergence may not imply the uniform convergence when X is an arbitrary (infinite) set.

One can observe that in the mathematical literature, there are very few known results that give conditions under which a pointwise convergence implies the uniform convergence. Concerning sequences of continuous functions defined on a compact set, we have the following facts:

Proposition A. (Dini’s Theorem) [

If K is a compact metric space,

Proposition B. [

If E is a Banach space and

(For the sake of completeness, we give the proof of this proposition in the Appendix Section).

Therefore our aim is to highlight a new basic criterion that shows in some way how a sequence of real-valued functions can converge uniformly when it is more or less obvious that the sequence converges uniformly away from a finite number of points of the closure of its domain. In the case of sequences of functions of a real variable, our criterion avoids, unlike in most classical textbooks [

Let

Suppose that there exists a function f from

Suppose furthermore that for each

Then the sequence of functions

Proof

Let

Thus

by the uniform convergence of

And so

i.e.,

The boundedness condition (D) of the above theorem can not be removed as shown by the sequence of functions defined from

where

And we can see that

We give some examples that illustrate the theorem.

(1) Let

Then the sequence of functions

converges uniformly to

(2) Given an infinite metric space

i) the sequence of functions

converges uniformly to 0 on E,

ii) the sequence of functions

converges uniformly to 0 on E.

(3) Let

i) Consider the sequence of functions

Then

ii) Consider the sequence of functions

Then

iii) Consider the sequence of functions

Then

(4) In real analysis, we can recover the facts that each of the following sequences converges uniformly to 0 on their respective domains:

Justifications (Proofs) of the examples

(1) For every

Therefore, on the one hand, for each

showing that

On the other hand, we have

fulfilling condition (D) of the above theorem.

Thus

(2) i) On the one hand, for each

and so

On the other hand, we have

fulfilling condition (D) of the above theorem.

Thus

ii) The uniform convergence of

Observe that the uniform convergence of

(3) Note that for all natural number n, we have

because

following from

Therefore it suffices to prove that

Let

Then on the one hand, for each

and so

On the other hand, we have

showing condition (D) of the above theorem.

Thus

(4) i) Let us set

On the one hand, we have for every

On the other hand, we have for every

showing that

Therefore, by taking

ii) For

On the one hand, we have for every

On the other hand, we have for every

showing that

Therefore, by taking

iii) For

On the one hand, we have for every

On the other hand, we have for every

showing that

Therefore, by taking

iv) For

On the one hand, we have for every

On the other hand, we have for every

showing that

Therefore, by taking

v) The example of

GuyDegla,11, (2015) A Remark on the Uniform Convergence of Some Sequences of Functions. Advances in Pure Mathematics,05,527-533. doi: 10.4236/apm.2015.59048

In this section, we prove Proposition B for the sake of completeness.

Proof of Proposition B

Let

Also,

It follows that