# Real Analysis/Uniform Convergence

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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 ${\displaystyle |f_{n}(x)-f(x)|<\epsilon }$
Let ${\displaystyle f_{n}}$ be a series of continuous functions that uniformly converges to a function ${\displaystyle f}$. Then ${\displaystyle f}$ is continuous.
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.