Question

If a cauchy sequence has a convergent subsequence, then prove that the whole sequence is convergent.

Answer 1

Call the sequence anan and its convergent subsequence ankank with limit aa. Then for every ε>0ε>0 there exists a N∈NN∈N such that |ank−a|<ε2|ank−a|<ε2 for nk>Nnk>N. Also as anan is a Cauchy sequence |ank−an|<ε2|ank−an|<ε2 for nk,n>N1nk,n>N1. Now choose N2N2 as the maximum of N1,NN1,N and you have by using the triangle inequality that: 
|an−a|=|an−ank+ank−a|≤|an−ank|+|an−a|=|an−ank+ank−a|≤|an−ank|+
 |ank−a|<ε2+ε2=ε|ank−a|<ε2+ε2=ε for all n,nk>N2n,nk>N2which is convergence by definition