# Cauchy and sequences

Not what you're looking for? Search our solutions OR ask your own Custom question.

See attachment

1. Let ( ) be a bounded sequence. Show that there exists a subsequence of ( ) converging to

2. Show that is not a Cauchy sequence Conclude that diverges.

Â© BrainMass Inc. brainmass.com November 30, 2021, 2:32 am ad1c9bdddfhttps://brainmass.com/math/algebra/cauchy-bounded-sequences-184438

#### Solution Preview

Please see the attachment.

Problem #1

Proof:

Let , then is a monotone increasing sequence. Since is a bounded sequence, then is bounded. Thus has a limit. So we have

We also have for any .

We have two cases:

Case 1: We can find some , such that for all ...

#### Solution Summary

This is a proof regarding subsequences of a bounded sequence and another proof regarding a Cauchy sequence.

$2.49