# Real Analysis/Uniform Convergence

 Real Analysis Uniform Convergence

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

### Theorem (Uniform Convergence Theorem))

Let ${\displaystyle f_{n}}$  be a series of continuous functions that uniformly converges to a function ${\displaystyle f}$ . Then ${\displaystyle f}$  is continuous.

#### Proof

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