Continuous spectrum is a subset of point spectrum

Question

I have to prove that the continuous spectrum $\sigma_c(T)$ is a subset of the point spectrum $\sigma_p(T)$.

I started off by supposing that there is some spectral value $\lambda$ such that $\lambda \notin \sigma_p(T)$.

Then, this would imply that $(T-\lambda I)$ is invertible.

How can i proceed further to show that $(T-\lambda I)$ is not bounded so that $\lambda$ is also not in $\sigma_c(T)$, thus proving that,$\sigma_c(T) \subset \sigma_p(T)$ ?

Answer 1

This is not true !! to convince you, take $T$ the unilateral shift so : $$ \sigma(T)=\bar{\mathbb{D}}\\ \sigma_p(T)=\mathbb{D}\\ \sigma_c(T)=\mathbb{T} $$ proof page 6-7.