i lost the way when i read the proof of Eckart–Young–Mirsky theorem (for Frobenius norm) recorded on wikipedia(enter link description here).
i firstly want to know in the following logitic of the proof, \begin{aligned}\sigma _{i}(A')+\sigma _{j}(A'')&=\sigma _{1}(A'-A'_{i-1})+\sigma _{1}(A''-A''_{j-1})\\&\geq \sigma _{1}(A-A'_{i-1}-A''_{j-1})\\&\geq \sigma _{1}(A-A_{i+j-2})\qquad ({\text{since }}{\rm {rank}}(A'_{i-1}+A''_{j-1})\leq {\rm {rank\,}}(A_{i+j-2}))\\&=\sigma _{i+j-1}(A).\end{aligned}
how to conclude $${\rm{rank}}(A'_{i-1}+A''_{j-1})\leq {\rm {rank\,}}(A_{i+j-2})$$ And my second question is why above conclusion can get
$$\sigma _{1}(A-A'_{i-1}-A''_{j-1})\geq \sigma _{1}(A-A_{i+j-2})$$
thank for u help in advance.