If $K$ and $L$ are nth syzygies modules for $M$ then $K\sim L$
$K\sim L$ means there exists P and Q are projective modules such that $K\oplus Q=L\oplus P$
I try to use induction so it suffices to show that if $K$ and $L$ are the first syzygies module for $M$ and $N$ where $M\sim N$ then $K\sim L$ but I got stuck there.