How to prove this problem?
Let X be a non-zero Banach space and $T \in \mathcal{L}(X).$ Let $\rho(T)$ be the resolvent set, $\sigma(T)$ the spectrum of T. Suppose $\{\lambda_n\}_{n=1}^\infty \subset \rho(T)$ and $\lambda \in \sigma(T)$ such that $\lambda_n \rightarrow \lambda.$ Prove that $\sup_n\|(\lambda_nI - T)^{-1}\|_{\mathcal{L}(X)} = \infty.$
I have tried to use this Norm of the Resolvent but operator is not self-adjoint. **