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

Compact. Let $P_N$ keep the first $N$ coordinates. The operator $P_{N}T$ has finite rank and $\Vert T-P_{N}T\Vert \le 1/(N+1)$, since the remaining diagonal coefficients are at most that number. Norm-closedness of compact operators proves the claim.