= Solution
Let $P_N$ be the orthogonal projection of $\ell^2$ onto the span of its first $N$ standard basis vectors. Then $P_Ny\to y$ for every $y\in\ell^2$, and $\lVert I-P_N\rVert\leq1$.
Let $T$ be compact and let
$$
K=\overline{T(B)},
$$
where $B$ is the closed unit ball. The set $K$ is compact. Given $\varepsilon>0$, choose a finite $\varepsilon/3$-net $y_1,\ldots,y_m$ in $K$. Pointwise convergence lets us choose $N$ such that
$$
\lVert(I-P_N)y_j\rVert<\frac{\varepsilon}{3}
$$
for every $j$. If $y\in K$ and $\lVert y-y_j\rVert<\varepsilon/3$, then
$$
\lVert(I-P_N)y\rVert
\leq\lVert(I-P_N)(y-y_j)\rVert
+\lVert(I-P_N)y_j\rVert
<\frac{2\varepsilon}{3}.
$$
Thus $P_N\to I$ uniformly on $K$, and
$$
\lVert P_NT-T\rVert
=\sup_{\lVert x\rVert\leq1}\lVert(P_N-I)Tx\rVert
\longrightarrow0.
$$
Each $P_NT$ has image in an $N$-dimensional space, so it has finite rank. This <coordinate-projection approximation of a compact operator> proves the result.
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