In connection with the question
I can't get satisfied with such 'so on' type logic. Is there a better way to solve it?
and the responses recieved I would like to know whether recursive relationship is a different version of principle of mathematical induction?
I can see here $x_1\in X$ and $x_n\in X\implies~\exists~x_{n+1}\in X$ such that $d(x,x_n)>d(x,x_{n+1}).$ Can we conclude by induction that $\exists~\{x_n\}\subset X$ such that $d(x,x_1)>d(x,x_2)>...?$