Show $f_n(x_n) \to f(x)$ if $f_n \to f$ uniformly and $x_n \to x$.
$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.