Real Analysis/Uniform Convergence

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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 $\epsilon >0$ there is an $N>0$ such that when $n>N$ for every x in the domain of the functions f, then $|f_{n}(x)-f(x)|<\epsilon$

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 $|f(x')-f(x)|\leq |f(x')-f_{n}(x')|+|f_{n}(x')-f_{n}(x)|+|f_{n}(x)-f(x)|<{\frac {\epsilon }{3}}+{\frac {\epsilon }{3}}+{\frac {\epsilon }{3}}=\epsilon$ so the function f(x) is continuous.