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