= Solution
The <spectral theorem for compact Hermitian operators> says that the nonzero spectrum of a compact Hermitian $T$ consists of real eigenvalues of finite multiplicity, with zero as the only possible accumulation point. Eigenvectors for distinct eigenvalues are orthogonal, and an orthonormal basis of $\ell^2$ can be chosen from eigenvectors together with a basis of $\ker T$. Thus
$$
Tx=\sum_{n\geq1}\lambda_n\langle x,e_n\rangle e_n,
$$
where the nonzero eigenvalues are repeated according to multiplicity and $\lambda_n\to0$ if there are infinitely many.
Define
$$
T_Nx=\sum_{n=1}^N\lambda_n\langle x,e_n\rangle e_n.
$$
Each $T_N$ is finite-rank and Hermitian, and
$$
\lVert T-T_N\rVert
=\sup_{n>N}|\lambda_n|\longrightarrow0.
$$
\b[Hence every compact Hermitian operator is a norm limit of finite-rank Hermitian operators, by <finite-rank truncation of a compact Hermitian operator>.]
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