I've read the proof of the Eckart-Young-Mirsky theorem on WikiPedia (for Frobenius norm), and I understand all of it except this one step:
$\sigma_1(A - A'_{i - 1} - A''_{j - 1}) \geq \sigma_1(A - A_{i + j - 2})$
Note that $A' + A'' = A$, and $A_{k}$ is the rank-k SVD-based approximation of a given matrix $A$ (i.e. $A_k = \sum_{i=1}^{k}\sigma_iu_iv_i^\top$). See the reference for more context if necessary.
The article justifies this step by stating that $rank(A'_{i-1} + A''_{j-1}) \leq rank(A_{i + j - 2})$; I think I understand why this rank inequality is true, but I don't see how it justifies the spectral norm inequality. What am I missing?