Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 11G / b / iii / 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 $d_m$, soThe conclusion depends on the point $x_m$ itself, even if that point appears at several indices.