Diagonalizable Operator Compact iff Value Set Converges to Zero

Theorem
Let $H$ be a Hilbert space of countable dimension.

Let $A: H \to H$ be a diagonalizable operator.

Let $\left({\alpha_n}\right)_{n \in \N}$ be the value set of $A$, with respect to a suitable basis $E = \left({e_n}\right)_{n \in \N}$ for $H$.

Then $A$ is compact iff:


 * $\displaystyle \lim_{n \to \infty} \alpha_n = 0$