Diagonalizable Operator Bounded iff Value Set Bounded

Theorem
Let $H$ be a Hilbert space.

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

Let $\family {\alpha_e}_{e \mathop \in E}$ be the value set of $A$, a suitable basis $E$ for $H$.

Then $A$ is bounded $\family {\alpha_e}_{e \mathop \in E}$ is bounded.

That is, :
 * $\exists M \in \R: \forall e \in E: \size {\alpha_e} \le M$