Show these operators converge to a particular limit

Question

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H and ${\lambda}_{n}$ are the corresponding eigenvalues. There are either finitely many ${\lambda}_{n}$ or they can be arranged in a sequence tending to zero. I want to show that $T$ is compact by showing it is the limit of the operators $T_{n}=\sum_{k=1}^{n}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ . In the case where the ${\lambda}_{n}\to0$, I have shown that $\|T-T_{n}\|$ is less than or equal to $\sup\{|{\lambda}_{k}| : k>n\}$ which tends to zero as $n\to\infty$ and hence I have shown convergence to $T$. But in the case where the ${\lambda}_{n}$ form a finite set, this argument doesn't work.