General Topology – Proving Lower Semicontinuous Function is Supremum of Increasing Sequence of Continuous Functions

analysisgeneral-topologysemicontinuous-functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.

My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.

Can anyone help?

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This is a quotation from "General Topology" by Ryszard Engelking:

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