Relation between $\epsilon$-pseudospectrum of operators

Question

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why

$\sigma_{\epsilon}(T)\subseteq \sigma_{\epsilon}(T+S)$?

where $\sigma_{\epsilon}(T)=\{\lambda\in\mathbb{C}:||(\lambda-T)^{-1}||\geq 1/\epsilon\}$ with this convention that if $T$ is not invertible, then $||T^{-1}||=\infty$.

Answer 1

I think this problem is Theorem 2.4(iii) of the book written by, L. N. Trefethen, M. Embree, spectra and pseudospectra, 2005.