 Solution

The set $S=\{x_n:x_n\ne \ell \}$ is countable, hence closed, and does not contain $\ell$. Its complement is an open neighborhood of $\ell$. Convergence therefore places every sufficiently late term outside $S$, which is possible only if those terms equal $\ell$. Thus the sequence is eventually constant at its limit, and the set of exceptional indices is finite. This conclusion concerns repeated indices as well as distinct values.