Let $f\_n$ be a sequence of functions, $f\_n : S\rightarrow T$, not necessarily continuous and suppose that $f\_n \rightarrow f$ as $n \rightarrow \infty$. Let $f$ be uniformly continuous. I.e. for all $\epsilon \gt 0, \exists \delta \gt 0$ such that $\forall x\in S, |f(x) - f(y)| \lt \epsilon$ whenever $|x - y| \lt \delta, y\in S$.

We want to show that for all $\epsilon \gt 0, \ \exists N$ such that for all $x \in S, n\gt N$, we have $|f\_n(x) - f(x)|\lt \epsilon$.

If we need it, $S$ is compact and $T = \mathbb{C}$.

Since $S$ is compact, by the extreme value theorem there exists $c,d \in S$ such that $f(c) \leq f(x) \leq f(d), \ \forall x \in S$.