Real Analysis/Uniform Convergence

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))

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.

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