Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact operators $(T_n)_{n=1}^\infty$ on $K$ such that $T_n\xi_n = 0$ and $\|T_n - T\|\to 0$ as $n\to \infty$?
This seems to be true but the obvious first choice, namely $P_{n-1}T$, where $P_n$ is the projection onto the span of $\{\xi_1, \ldots, \xi_n\}$ does not work.
Notice that $T\xi_n \xrightarrow{n\to\infty} T\xi = 0$, so $\lVert T\xi_n \rVert \to 0$.
Let $$ S_n(x) = \begin{cases} \frac{\langle Tx,T\xi_n\rangle}{\lVert T\xi_n \rVert^2} T\xi_n & \text{if } T\xi_n \neq 0 \\ 0 & \text{if } T\xi_n = 0\end{cases} $$ Then $S_n$ is compact, $S_n(\xi_n) = T\xi_n$, and $\lVert S_n \rVert \to 0$, so $T_n = T - S_n$ does what you want.