I have to show that for a self-adjoint operator $(A,D(A))$ on a seperable Hilbert space $\mathcal{H}$ and $f\colon \mathbb{R}\to \mathbb{R}$ continuous and bounded that
\begin{align*} \sigma(f(A))=\overline{f(\sigma(A))}\, . \end{align*}
I thought about using the multiplication operator version of spectral theorem, but I don't quite know how to start... I would really like to do the proof on my own, but I would be really greatful for a tip! Thanks!