I am studying some properties of compact normal operators. I have found that if $T$ is a compact normal operator on an infinite dimensional Hilbert space $\mathcal H$ then $\sigma (T) = \sigma_p(T) \cup \{0\}.$ Also since $0$ is the only possible limit point of $\sigma (T)$ it turns out that $\sigma_p(T)$ would be countable otherwise $\sigma_p(T)$ would have a limit point in it, a contradiction. So if $\sigma (T)$ is infinite then it is of the form $\sigma (T) = \{\lambda_i\ |\ i \in \mathbb N \} \cup \{0\},$ where $\lambda_i$'s are the eigenvalues of $T.$ Also I came across the fact that $T = \sum\limits_{j = 1}^{\infty} \lambda_j P_j,$ where $P_j$'s are the pairwise orthogonal projections and the infinite sum on the right converges to $T$ in norm.
Now in view of the above facts our instructor also claimed that $\lim\limits_{j \to \infty} \lambda_j = 0$ and $P_j$'s are all finite rank projections since $T$ is compact. But I can't get his point. Could anyone please clear it to me?
Thanks for your time.