I need to prove that the resolvent set of the operator $T$ , is an open set. I saw the proof in the book of Bernard Beauzamy and I understand the work done there except one part.
The proposition states that the resolvent set of an operator is an open set.
Proof: We fix $\lambda \in \rho(T)$ and let $\varepsilon \in \mathbb{C}$, with $|\varepsilon|<\frac{1}{||R(\lambda)||}$. We will show that $\lambda +\varepsilon \in \rho(T)$.
I do not see why $\lambda +\varepsilon \in \rho(T)$ implies that $\rho(T)$ is open.
Thank you for helping!