**Definition. **

We call a sequence a Cauchy sequence if for all there exists an such that for all m, .

**Theorem:**

Every Cauchy sequence is a bounded sequence and is convergent.

**Proof.**

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 .

**QED.**

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**).

**Theorem:**

For sequences and .

(i) If and , then too is convergent and .

(ii) If , then , .

(iii)

(iv)

(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.

More later,

Nalin Pithwa