Real Analysis/Uniform Convergence

Real Analysis
Uniform Convergence

Definition: A sequence of real-valued functions is uniformly convergent if there is a function f(x) such that for every there is an such that when for every x in the domain of the functions f, then

Theorem (Uniform Convergence Theorem)) edit

Let   be a series of continuous functions that uniformly converges to a function  . Then   is continuous.

Proof edit

There exists an N such that for all n>N,   for any x. Now let n>N, and consider the continuous function  . Since it is continuous, there exists a   such that if  , then  . Then   so the function f(x) is continuous.