Lower semicontinuous approximation

2018-03-12 20:23:22

Suppose $u : X \to \mathbb{R} \cup \{+\infty\}$ is a lower semicontinuous function on a topological space $X$. Suppose $X$ compact and Hausdorff. Prove that there exists a non decreasing sequence of continuous functions pointwise converging to $u$.

You can suppose that $X$ has a countable basis, if you need it, but I don't think you need it. I know a proof in the context of metric spaces, but I want a detailed proof in the case of $X$ Hausdorff and compact.