Suppose {pn} is a cauchy sequence in a metric space X and some subsequence {pnk} converges to a point a belongs to X.
Let ϵ>0. There exists N such that, for n,m≥N,d(pn,pm)<ϵ ( {pn} is a cauchy sequence ). Then, consider any n≥N :
for k sufficiently large (so that nk≥N, ), d(pn,pnk)≤ϵ .
Taking the limit as k→∞, it follows that d(pn,a)≤ϵ .
Or: if k is sufficiently large then d(pnk,a)<ϵ (by the assumption pnk→a ), so d(pn,a)≤d(pn,pnk)+d(pnk,a)<2ϵ.
In any case, we conclude that d(pn,a) becomes arbitrarily small for n large, i.e. pn→a .
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