Past exam of the functional Analysis and Banach and C*-Algebras course of the University of Oxford / 2007 / A / 4 / iii / Solution

Extract a subsequence whose $T$ images converge, and apply continuity of the bounded operator $S$ to those images. This proves $ST$ is compact. The reverse composition $TS$ is compact as well, since $S$ sends a bounded sequence to a bounded sequence to which compactness of $T$ applies.