# $\sigma_p(T)$ is at most countable for a normal operator

This is the part (b) of Rudin's Functional Analysis Chapter 12, question 18.

Show that if $H$ is a separable Hilbert space then the point spectrum (denoted by $\sigma_p(T)$) of a normal operator $T$ is at most countable.

I have done part (a) and (c)
Part (a) is to prove that, the residual spectrum $\sigma_r(T)$ of a normal $T \in \mathcal{B}(H)$ is empty. (perhaps this result is useful)