Let $T$ be a compact operator.Then show that $\sum_{n=1}^\infty$$M_n(T)=\sup\{||PT||_1:P\mbox{ is a rank } N\mbox{ projection}\}$ where $M_n$'s are the singular values of $T$ and $||A||_1=\sum_{n=1}^\infty M_n(A)$.
I am trying to prove this result by using the fact that $M_i(PT)$||$<$||$P$||$M_i(T)$ but am unable to get the final result.Thanks for any help.