Let $\|\cdot\|$ be a unitary invariant norm on $\mathbb{R}^{m \times n}$. $\forall A \in \mathbb{R}^{m \times n}$, suppose the SVD of $A$ is $$A = \sum_{i=1}^{r} \sigma_i u_i v_i^{\mathrm{T}}$$ where $r = \operatorname{rank}(A)$. For $k \leq r$, prove that $$A_k = \sum_{i=1}^{k} \sigma_i u_i v_i^{\mathrm{T}}$$ is the best rank-$k$ approximation of $A$.
It is known that the conclusion holds for $\|\cdot\|_F$ and $\|\cdot\|_2$. The proofs can be found in https://en.m.wikipedia.org/wiki/Low-rank_approximation. But how to prove it for general unitary invariant norm?
Any suggestion is appreciated. Thanks!