# Real Analysis/Uniform Convergence

 Real Analysis Uniform Convergence

## Contents

Definition: a sequence of real-valued functions fn(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 |fn(x)-f(x)|<ε

### Theorem (Uniform Convergence Theorem))

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

#### Proof

There exists an N such that for all n>N, $|f_n(x) - f(x)|<\frac{\epsilon}{3}$ for any x. Now let n>N, and consider the continuous function $f_n$. Since it is continuous, there exists a $\delta$ such that if $|x'-x|<\delta$, then $|f_n(x)-f_n(x')|<\frac{\epsilon}{3}$. Then $|f(x')-f(x)|\le |f(x')-f_n(x')|+|f_n(x')-f_n(x)|+ |f_n(x)-f(x)| < \frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3} = \epsilon$ so the function f(x) is continuous.