Question #91437
Suppose {p} is a cauchy sequence in a metric space X and some subsequence {p} converges to a point a belongs to X . Prove that the full sequence {p} converges to a .
1
Expert's answer
2019-07-05T11:01:33-0400

Suppose {pn}\{{p_n} \} is a cauchy sequence in a metric space X and some subsequence {pnk}\{p_{n_k}\} converges to a point aa belongs to XX.

Let ϵ>0.\epsilon > 0. There exists NN such that, for n,mN,d(pn,pm)<ϵn, m \geq N , d(p_n, p_m) <\epsilon ( {pn}\{{p_n} \} is a cauchy sequence ). Then, consider any nNn \geq N :

for kk sufficiently large (so that nkN,n_k \geq N, ), d(pn,pnk)ϵd(p_n, p_{n_k} ) \leq \epsilon .

Taking the limit as kk \rightarrow \infty, it follows that d(pn,a)ϵd(p_n, a)\leq \epsilon .

Or: if kk is sufficiently large then d(pnk,a)<ϵd(p_{n_k}, a) <\epsilon (by the assumption pnkap_{n_k }\rightarrow a ), so d(pn,a)d(pn,pnk)+d(pnk,a)<2ϵd(p_n, a) \leq d(p_n, p_{n_k}) + d(p_{n_k}, a) < 2 \epsilon.

In any case, we conclude that d(pn,a)d(p_n, a) becomes arbitrarily small for nn large, i.e. pnap_n \rightarrow a .


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