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.