# Real Analysis/Uniform Convergence

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.