Three things you need to know. (1) Weakly convergent sequences are bounded. (This follows from the uniform boundedness principle, because a weakly convergent sequence is obviously weakly bounded.) (2) Compact operators convert weakly convergent sequences into norm-convergent sequences. (3) The set of compact operators is norm-closed, so that if C_n→C_0 in norm and each C_n is compact, then so is C.

We want to show that . Here's an outline of how to do it. First, for n large enough, Cx_n is close to Cx_0 in norm, by (2) and (3). Next, for n large enough, C_nx_n is close to Cx_n (this follows from (1), together with the fact that C_n→C_0 in norm.

Putting those two statements together, you see that (for n large enough) is close to . But the weak convergence tells you that (again for n large enough) is close to . Therefore as n→∞.