Question: Prove that a sequence of real numbers is convergent if and only if it is a Cauchy sequence.
I'm currently learning real analysis through an inquiry based course, and I'm trying to prove the above statement in the backwards direction. I've already proved that every Cauchy sequence is bounded (using similar logic to this proof), so now I'm trying to see how I can use that information in my proof.
Most proofs that I have seen across the internet use the "Bolzano-Weierstrass theorem," which is something that is not in the text (and it seems like a pretty involved proof), so I'm trying to see if there's another way to complete this proof.
We are allowed to assume that a monotone sequence is convergent iff it is bounded, but the text doesn't say much about monotone sequences, so I'm not sure if that information is helpful or not.
Thanks for any help you can give me in understanding this concept. I'm happy to elaborate where I can.