For a square matrix $A \in \mathbb{C}^{n \times n}$ with the singular value decomposition $A = U\Sigma V^*$, I want to show that
$$\|A - P \|_{2} \leq \|A -W \|_{2}$$
Where $P = UV^{*}$ and $W$ is an arbitrary unitary matrix.
It is immediately clear to me that
$$\|A - P \|_{2} = \|U\Sigma V^* - UV^{*}\|_2 = \|\Sigma - I \|_2 $$
I also know that the singular values of all unitary matrices are all one. However, I don't know how to combine this fact with properties of the spectral norm to get a proof, if this is indeed the right way.