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.
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 CaptainBlank for teaching this move to me).
We have,
for .
Q.E.D.
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.