Uniformly Convergent Series of Continuous Functions Converges to Continuous Function

Theorem
Let $S \subseteq \R$.

Let $x \in S$.

Let $\sequence {f_n}$ be a sequence of real functions.

Let $f_n$ be continuous at $x$ for all $n \in \N$.

Let the infinite series:


 * $\displaystyle \sum_{n \mathop = 1}^\infty f_n$

be uniformly convergent to a real function $f : S \to \R$.

Then $f$ is continuous at $x$.

Proof
Let $\sequence {s_n}$ be sequence of real functions $S \to \R$ with:


 * $\displaystyle \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$

for each $n \in \N$ and $x \in S$.

By Combination Theorem for Continuous Functions: Sum Rule:


 * $s_n$ is continuous at $x$ for all $n \in \N$.

Since additionally $s_n \to f$ uniformly, we have by Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function:


 * $f$ is continuous at $x$.