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 $δ$ such that if $| x^{'} - x | < δ$ , then $| f_{n} (x) - f_{n} (x^{'}) | < \frac{ϵ}{3}$ . Then $| f (x^{'}) - f (x) | \leq | f (x^{'}) - f_{n} (x^{'}) | + | f_{n} (x^{'}) - f_{n} (x) | + | f_{n} (x) - f (x) | < \frac{ϵ}{3} + \frac{ϵ}{3} + \frac{ϵ}{3} = ϵ$ so the function f(x) is continuous.