An unbounded sequence containing a subsequence that is Cauchy.

Answer 1

Sorry, what in the question here? This is certainly not true generically. Are you looking for necessary and sufficient conditions, or just an example? If you just want an example then you could take the sequence a_n = (n)^((-1)^n). This is unbounded but contains the Cauchy subsequence 1/1, 1/3, 1/5, ...