Spectral perturbation of self-adjoint operators in Hilbert space

Question

Let $H$ be a Hilbert space, $A:H\rightarrow H$ be a bounded self-adjoint operator, then the spectrum $$\sigma(A+\varepsilon I)=\sigma(A)+\varepsilon,$$ Now suppose $B:H\rightarrow H$ is another bounded self-adjoint operator, what can we say about the spectrum $\sigma(A+\varepsilon B)$? In particular, I concern that

1. Is it right that $\sigma (A+\varepsilon B)\rightarrow \sigma (A)$ as $\varepsilon \rightarrow 0$? In what sense?
2. Suppose item 1 is right, what can we say about the convergence rate?

I have no idea about solving this, could you offer me some references or partial answer? Any help will be appreciated, thank you!