When talking about non convergence of Cauchy sequences, the only examples I can found are the ones in which the sequences converges outside the space (X,d), ans so it does not converge in X.
Can I have a Cauchy sequence that is not convergent not because it's limits falls outside the space but because it is $+-\infty $ or because it does not exist like $ (-1)^n$?
I already ruled ot the first case, since every Cauchy sequence is bounded, so I still need to prove if the limit always exist or give a counterexample of a non-existent limit. How do I go about it?