Past exam of the options I course of the University of Oxford / 2007 / E / E1 / a / Solution

A sequence converges if there is $x\in X$ such that, for every $ϵ>0$, some $N$ satisfies $d(x_n,x)<ϵ$ whenever $n\ge N$. The limit must belong to $X$, not merely to a larger ambient space.