If a sequence is increasing and has a Cauchy subsequence, is the original sequence Cauchy?

Question

I am having difficulty answering whether an increasing sequence with a Cauchy subsequence implies that the original sequence is Cauchy, and how I would go about showing that using some key analysis theorems and logic.

Answer 1

Hint: By relating the general indices and the indices of the subsequence, try to bind the distance between an arbitrary term and a term of the Cauchy subsequence.

Answer 2

HINT: You know that the Cauchy subsequence converges; let $a$ be its limit.

• Show that $a$ is an upper bound for the original sequence. (This is not strictly speaking necessary, but doing it may give you a better feel for what’s going on.)
• Show that the original sequence converges to $a$. Then use the fact that every convergent sequence is Cauchy.