Let $(X,d)$ be a metric space and $\{x_n\}_n$ a sequence. I'm looking for an algorithm that takes $x\in X$ as an input and gives a $y\in X$ as an output, with this property: if $\{x_n\}_n$ admits some converging subsequence then $\{y_n\}_n$ is a converging subsequence of $\{x_n\}_n$.
I started naively with this: let $y_0=x_0$, then take $y_1=x_{i_1}$ where $x_{i_1}$ is the first such that $d(x_0,x_{i_1})<2^{-1}$, if this exists, and $y_1=y_0$ otherwise, and so on. But this doesn't work, since a converging subsequence could easly start at some $n>0$ and i will not find it in this way.
Have someone any idea? Is there such an algorithm?
Remark: this problem could be setting in a two player, perfect information game-context: player I enumerate a sequence by playing $x\in X$ at each of its turn and player II has to choose if take or not the $x$ II just played, in order to have, at the end of the game, a subsequence of $\{y_n\}_n$ such that if $\{x_n\}_n$ (the II's run) admits some converging subsequence, then $\{y_n\}_n$ is such a subsequence.