Let $X$ be compact and Hausdorff, and let $\mathcal{B}(X)$ be the set of bounded Borel-measurable functions $X \to \mathbb{C}$. Also let $\mathcal{C}(X)$ be the set of continuous functions $X \to \mathbb{C}$.
In my notes from class, I wrote that for any $h \in \mathcal{B}(\sigma(A))$ there exists a sequence $(g_n) \subset \mathcal{C}(\sigma(A))$ such that $(g_n) \overset{\Vert \cdot \Vert_{\infty}}{\longrightarrow}h$. Why is this true?
This is part of the development of the Borel functional calculus, so $\sigma(A)$ is the spectrum of a self-adjoint linear operator $A \in \mathcal{L}(\mathcal{H})$.