A Remark on the Uniform Convergence of Some Sequences of Functions

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.


Introduction
Let X be a nonempty set, : f X →  be a function and { } n n f ∈ be a sequence of real-valued functions from X into  .Recall [1]- [3] that the sequence { } f ∈ converges uniformly to f on X, then for each x X ∈ fixed, the sequence f ∈ converges pointwise to f.It is also obvious that when X is finite and

{ }
n n f ∈ converges pointwise to f on X, then { } n n f ∈ converges uniformly to f on X. However this converse 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) [4] If K is a compact metric space, : T ∈ is a sequence of bounded linear operators of E that converges pointwise to a bounded linear operator T of E, then for every compact set K E T ∈ converges uniformly to T on K.
(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 [3] [6], the search of extrema (by differential calculus) of their general terms.Several examples that satisfy the criterion are given.

Theorem
Let ( ) , E d be a metric space and Ω ≠ ∅ be a subset of E. Consider a sequence { } n n f ∈ of functions defined from Ω to  .
Suppose that there exists a function f from Ω to  , some points  ε > be arbitrarily fixed (it may be sufficiently small in order to be meaningful).Then for every natural number n, we have And so ( ) ( )

Observation
The boundedness condition (D) of the above theorem can not be removed as shown by the sequence of functions defined from [ ] 0,1 into  as follows: where  is equipped with its standard metric.Indeed, { } ε ∈ , but with 1 k = and 1 0 a = there is no positive number r for which the condition (D) is satisfied since ( )

Examples
We give some examples that illustrate the theorem.
(1) Let ( ) , E d be an infinite metric space and let a E ∈ be fixed.Denote by ϕ the function defined from E into  by , (2) Given an infinite metric space ( ) , converges uniformly to 0 on E.
(3) Let ( ) , E d be an infinite metric space and Ω be a bounded and infinite subset of E, let a and b be two different points of Ω and let α and β be two fixed positive numbers.i) Consider the sequence of functions { } g ∈ converges uniformly to 0 on Ω .
iii) Consider the sequence of functions { } h ∈ converges uniformly to 0 on Ω .
(4) In real analysis, we can recover the facts that each of the following sequences converges uniformly to 0 on their respective domains: ( ) ( ) , Therefore, on the one hand, for each 0 ε > , we have showing that { } (2) i) On the one hand, for each 0 ε > , we have for all ( ) On the other hand, we have fulfilling condition (D) of the above theorem.
Thus { } n n u ∈ converges uniformly to 0 on E.
ii) The uniform convergence of { } and .
Observe that the uniform convergence of { } n n v ∈ could also be proved using directly the above theorem.(3) Note that for all natural number n, we have f ∈ converges uniformly to 0 on Ω , although each of these three sequences can be handled directly with the above theorem.
Let δ be the diameter of Ω .Then on the one hand, for each 0 ε > , we have for all On the other hand, we have showing condition (D) of the above theorem.
Thus { } n n f ∈ converges uniformly to 0 on Ω and we are done.
 .On the one hand, we have for every n ∈  : On the other hand, we have for every 1 0, 2 ψ converges uniformly to 0 on ( ) ii) For ( ) ( )  .On the one hand, we have for every n ∈  : On the other hand, we have for every 1 0, 2 ψ converges uniformly to 0 on ( )  .On the one hand, we have for every n ∈  : On the other hand, we have for every π 0, 4 Therefore, by taking E =  , π 0, 2  .On the one hand, we have for every n ∈  : On the other hand, we have for every π 0, 4 showing that { }

f
∈ is said to converge uniformly to f on X, if denotes the open ball of E centered at i a and with radius ε .Then the sequence of functions { } n n f ∈ converges uniformly to f on Ω .Proof Let 0 converges uniformly to ϕ on E.