I have the following problem:

Let X be a Hilbert space with the inner product <.|.>

Let $\displaystyle C_n \in C(X)$ be a family of compact operators that converges in the norm to operator C.

Let $\displaystyle (x_n)_n$ be a sequence that converges weakly to $\displaystyle x_0$

Does <$\displaystyle C_n x_n | x_n$> converge? If it does, where to?

I'm completely lost, please help.