Suppose $T_n$ are compact and $\|T_n-T\|\to0$. For any $\varepsilon>0$ take $n$ with $\|T_n-T\|<\varepsilon$. A finite $\varepsilon$-net of $T_n$ of the <unit ball> is then a finite $2\varepsilon$-net of its $T$ image. That image is <totally bounded>, so $T$ is compact by the stated criterion. Hence \b[$\mathcal K(X)$ is <norm> closed in $B(X)$].

Solution

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

 Solution (source code)