Let $A$ be a self-adjoint (not necessarily bounded) operator on a Hilbert space $\mathscr{H}$ and $B$ be self-adjoint bounded. Then how would one show that $\sigma (A+B) \subseteq \sigma(A) +\sigma(B)$? It seems that we would need a Neumann series argument, but I'm not sure of the details.
EDIT: @Jake28 Thanks for the counterexample. In that case, my question should be that how would you prove that $\sigma(A+B) \subseteq \sigma(A) + [-||B||,||B||]$? This was the original statement, but I though it would've been possible to extend it a little.
It might very well be that I misunderstand your question but if "self-adjoint bounded" means "self adjoint and bounded" then I think you need more assumptions on $A$ and $B$. If we consider the Hilbert space $\mathbb{R}^2$ and the operators \begin{align}A= \begin{pmatrix} 0 &1\\ 1&0 \end{pmatrix}, \qquad B=\begin{pmatrix} 1 &0\\ 0&0 \end{pmatrix} \end{align} then it's easy to check that $\sigma(A)=\{\pm1\}$ and $\sigma(B)=\{1,0\}$ whereas $\sigma(A+B)=\{\frac{1\pm \sqrt{5}}{2}\}$