Uniformly Convergent Sequence 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 $S \to \R$ converging uniformly to $f : S \to \R$.

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

Then $f$ is continuous at $x$.

Proof
Let $\varepsilon \in \R_{> 0}$.

Since $f_n \to f$ uniformly, there exists some $N \in \N$ such that:


 * $\size {\map {f_n} x - \map f x} < \dfrac \varepsilon 3$

for all $x \in S$ and $n \ge N$.

Since $f_N$ is continuous at $x$, there exists some $\delta > 0$ such that:


 * for all $y$ with $\size {x - y} < \delta$, we have $\size {\map {f_N} x - \map {f_N} y} < \dfrac \varepsilon 3$

Then for $y$ with $\size {x - y} < \delta$ we have:

Since $\varepsilon$ was arbitrary, $f$ is continuous at $x$.