In these notes, the following claim is made:
Let $H$ be self-adjoint and $A$ bounded then the spectrum of
$$T=H+A$$ is contained in
$$\sigma(T)\subset\{ \lambda; d(\lambda, \sigma(H))\le \Vert A \vert \}.$$
However, I could find no other reference for that statement, so I wonder whether it is true and if so, where I can find a proof. I should say that the non-trivial aspect of this statement is really that $A$ is not assumed to be symmetric, too.