In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim about the spectral mapping theorem's possible proof. Let me attempt to bring the context here. I should mention there are some nice results in this paper that I wanted to use and generalize for my own research, I hope to accurately bring the context below.
They bring up the continuos functional calculus $\phi: C(\sigma(A)) \rightarrow L(H)$ for a bounded, self-adjoint operator on a Hilbert space A. This is an algebraic *-homomorphism from the continuous functions on the spectrum of $A$ to the bounded operators on $H$. The paper's spectral mapping theorem basically says in this context $$ \sigma(\phi(f)) =f(\sigma(A)) $$ and the paper says something nice about this. It does not actually give a proof but it says there is a nice way to prove it using both inclusions with the inclusion $ f(\sigma(A)) \subseteq \sigma(\phi(f)) $ sketched in the following way: the author supposes $ \lambda \in f(\sigma(A)) $ and says "it is very obvious" that there exists a vector $h \in H$ with $\|h\|=1$ such that $\|\phi(f)-\lambda)h\|$ is arbitrarily small which shows $\lambda \in \sigma(\phi(f))$ which shows the desired inclusion.
The author says that it is "very obvious" to show this but I am a bit stumped. The way I would construct the continuous functional calculus is to start with polynomials and then generalize to $ C(\sigma(A)) $ based on the Weierstrass approximation theorem on the real compact set $\sigma(A)$ and the BLT theorem. The inclusion $\sigma(\phi(f)) \subseteq f(\sigma(A))$ is, I think, quite obvious but the other one in the above context has me stumped. Since I am already working on generalizing some results, I would really love to know how the author proves the inclusion with the method of showing the mentioned vector exists. Maybe use approximation in some way, but even though I suspect it is simple, I still do not see the author's proposed proof. Can someone here please help me recover it? I thank all interested persons.