An upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a topological space $X$ is always the pointwise infimum of a family of continuous functions $X \to \mathbb{R}$, and if $X$ is metric then the family of functions may be taken to be countable. This is in chapter 9 of Bourbaki's General Topology. Your result follows by the monotone convergence theorem.

For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\subset C(X,\mathbb{R})$ such that $f(x)=\inf\{g(x)\colon g \in \mathcal{F}\}$ for every $x \in X$. If $X$ is metrisable then $\mathcal{F}$ may be taken to be countable. These results can be found in chapter 9 of Bourbaki's General Topology. Your result follows by taking $(g_n)$ to be an enumeration of $\mathcal{F}$, defining $f_n:=\min_{1\leq k \leq n}g_k$, and applying the monotone convergence theorem to the sequence $(f_n)$ which decreases pointwise to $f$.