We call a sequence a Cauchy sequence if for all there exists an such that for all m, .
Every Cauchy sequence is a bounded sequence and is convergent.
By definition, for all there is an such that
for all m, .
So, in particular, for all , that is,
for all .
Let and .
It is clear that , for all .
We now prove that such a sequence is convergent. Let and . Since any Cauchy sequence is bounded,
But since is Cauchy, for every there is an such that
for all .
which implies that . Thus, for all . This is possible only if .
Thus, we have established that the Cauchy criterion is both a necessary and sufficient criterion of convergence of a sequence. We state a few more results without proofs (exercises).
For sequences and .
(i) If and , then too is convergent and .
(ii) If , then , .
(v) If and are both convergent, then , , and are convergent and we have , and .
(vi) If , are convergent and , then is convergent and .
Reference: Understanding Mathematics by Sinha, Karandikar et al. I have used this reference for all the previous articles on series and sequences.