$T$ is compact iff $\langle Te_i,e_j\rangle\to 0$ as $i,j\to\infty$

Question

Let $H$ be a separable Hilbert space and $T:H\to H$ be a bounded operator. Suppose $(e_i)$ is a countable orthonormal basis in $H$. Show that $T$ is compact if and only if $\langle Te_i,e_j\rangle\to 0$ as $i,j\to\infty$.

Note: I could prove that if $T$ is compact then $\langle Te_i,e_j\rangle\to 0$. I am stuck at the converse part.