I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎\_{‎ap‎}‎(T)‎‎‎‎‎$‎

Its proof is obvious from one hand.But i cant prove that $‎\sigma(T)\subseteq‎‎‎‎‎\sigma‎‎\_{‎ap‎}‎(T)‎‎‎‎‎$‎

Recall that $\sigma‎‎\_{‎ap‎}‎(T)‎‎‎‎‎=\{‎\lambda \in\mathbb{C}~|~T-\lambda~is~not~bounded~below‎‎\}‎$