I have the following question/task:
Let $X$ be a $C^{*}$-algebra, and let $x \in X$ be self-adjoint. Show that the following holds: $$\lVert (x - \lambda)^{-1}\rVert = \frac{1}{\text{dist}(\lambda, \sigma(x))}$$
where $\text{dist}(x, M)$ is defined as $\text{dist}(x,M) = \inf_{y \in M}\lVert x - y\rVert$.
We use the convention that $\frac{1}{0} = \infty$ and $\lVert (x - \lambda)^{-1}\rVert = \infty$ if $x - \lambda$ is not invertible.
My current approach: We have
$$\text{dist}(\lambda, \sigma(x)) = \inf_{\mu \in \sigma(x)} \lVert \lambda - \mu\rVert \Leftrightarrow \\ \frac{1}{\text{dist}(\lambda, \sigma(x))} = \frac{1}{\inf_{\mu \in \sigma(x)}\lVert \lambda - \mu\rVert} = \sup_{\mu \in \sigma(x)}\frac{1}{\lVert \lambda - \mu\rVert} = \sup_{\mu \in \sigma(x)} \lVert (\lambda - \mu)^{-1}\rVert$$
However, at this point, I am unsure how to proceed. Does anyone here have an idea on how to complete the proof? Also, where do I use the fact that $x \in X$ is self-adjoint? I'm not sure how to incorporate that into my proof.