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.© BrainMass Inc. brainmass.com November 30, 2021, 2:32 am ad1c9bdddf
Please see the attachment.
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 ...
This is a proof regarding subsequences of a bounded sequence and another proof regarding a Cauchy sequence.