# Real Analysis/Uniform Convergence

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Definition: a sequence of real-valued functions f_{n}(x) is uniformly convergent if there is a function f(x) such that for every ε>0 there is an N>0 such that when n>N for every x in the domain of the functions f, that |f_{n}(x)-f(x)|<ε

### 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.