Cauchy sequence w/ unbounded subsequence.
Is this possible? If so give an example, otherwise argue how it would be impossible?
I think there's different answers to this question depending on whether you are in a metric space but I'll assume not...
Here's a theorem for you.
Any Cauchy sequence is bounded.
Ill post up a proof if you need me to but basically just use the definition of a Cauchy sequence to show it converges.
Since any Cauchy sequence is bounded, its subsequences must be bounded.