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, then $|f_n(x)-f(x)|<ϵ$

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, $|f_n(x)-f(x)|<\frac{ϵ}{3}$ 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 $|x^{\prime }-x|<\delta$, then $|f_n(x)-f_n(x^{\prime })|<\frac{ϵ}{3}$. Then $|f(x^{\prime })-f(x)|\le |f(x^{\prime })-f_n(x^{\prime })|+|f_n(x^{\prime })-f_n(x)|+|f_n(x)-f(x)|<\frac{ϵ}{3}+\frac{ϵ}{3}+\frac{ϵ}{3}=ϵ$ so the function f(x) is continuous.