Every Convergent Sequence is Cauchy

Proof:

Let be a Convergent Sequence. Then , then (Limits of a Sequence > ^b9e295 )Let . Then by the Triangle Inequality

Every Cauchy Sequence is Convergent

Proof:

Step 1.
Every Cauchy sequence is bounded. This will allow us to apply the Bolzano Weismann Theorem > Bolzano Weismann Theorem to conclude that has a convergent subsequence .

Step 2.
If is a Cauchy sequence with a subsequence converging to , then also converges to .

Every Cauchy Sequence in is bounded.

Proof

If is Cauchy. For

Then

If we take
Then

If a Cauchy Sequence has a convergent subsequence, then it converges

For

Since

Note:

Hence Cauchy Sequences are Convergent

Remark: We say that a set is complete if every Cauchy sequence in converges to a point in . So, the above theorem says that is complete.