Let $(x_n)_{n \in \mathbb{n}}$ be a Cauchy sequence in some metric space $(X,d)$. I'm looking for different strategies that one could try to show that $(x_n)_{n \in \mathbb{N}}$ is convergent.
For example it is a well known that a Cauchy sequence in a metric space is convergent if and only if it possesses a convergent subsequence.
Are there similar results?
What about the special case of geodesic spaces? The space that I have in mind is not locally compact and thus Hopf-Rinow-Cohn-Vossen is unfortunately not applicable.