I know that (an) and (bn) are cauchy sequences. I need to prove or disprove that (an+bn) is also cauchy.

Thanks for any suggestions.

Printable View

- October 23rd 2006, 02:03 PMJaysFan31Cauchy sequence
I know that (an) and (bn) are cauchy sequences. I need to prove or disprove that (an+bn) is also cauchy.

Thanks for any suggestions. - October 23rd 2006, 05:48 PMThePerfectHacker
The sequence, is a Cauchy sequence. Thus, for we have,

for any

The sequence, is a Cauchy sequence. Thus, for we have, for any

Consider the sequence, . For any we can find satisfy those two conditions on top. Let then,

and,

.

Addition yields,

Triangular inequality (Thanks to Captain**Blank**for teaching this move to me).

We have,

for .

Q.E.D. - October 24th 2006, 12:03 AMCaptainBlack
- October 24th 2006, 06:59 AMThePerfectHacker
- October 24th 2006, 11:54 AMJaysFan31
Using the same from above, is (an*bn) Cauchy?

- October 24th 2006, 02:03 PMThePerfectHacker
Actually I have a much simpler proof that I should have stated for both the problems.

A sequence is a Cauchy sequence if and only if it is convergent.

Using this fact we can easily show that an*bn and an+bn is a cauchy sequence.

If an and bn are Cauchy sequences then they are convergent sequences. Then an*bn and an+bn are convergent sequences. That means then an*bn and an+bn are Cauchy sequences.