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

Given a bounded sequence, first pass to a subsequence on which the images under $T_1$ converge. From that subsequence extract another on which the images under $T_2$ converge. Both image sequences, and hence their sum, converge on the latter subsequence. Thus $T_1+T_2$ is compact.