Eigenspace for Normal Operator is Reducing Subspace

Theorem
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A \in B \left({H}\right)$ be a normal operator.

Let $\lambda \in \Bbb F$.

Then $\operatorname{ker} \left({A - \lambda}\right)$ is a reducing subspace for $A$.

Here $\operatorname{ker}$ denotes kernel.

Lemma
$A$ normal $\implies$ $\operatorname{ran}A \subseteq \operatorname{ker}(A)^\perp$

Corollary
$\operatorname{ran}(A-\lambda) \subseteq \operatorname{ker}(A-\lambda)^\perp$

Proof of corollary
Substitute $A-\lambda$ (which is also normal) for $A$.