Solution
= Solution
Write the <subsequence> as $y_n=x_{k_n}$ with strictly increasing indices $k_n$. If $y_\ell=x_m$, then $d(y_\ell,y_n)=d(x_m,x_{k_n})$. This is a <subsequence> of the real sequence converging to $d_m$, so
$$
\boxed{d(y_\ell,y_n)\longrightarrow d_m.}
$$
The conclusion depends on the point $x_m$ itself, even if that point appears at several indices.