We can say that any Cauchy sequence converges to some value in some space?

Question

Maybe this is a trivial question but I need to ask and clarify myself:

I know that any Cauchy sequence converges to some value inside some space if this space is complete.

But, we can say that any Cauchy sequences converges to some value to some space (not necessarily the space where the sequence is defined)?

In other words: exist a Cauchy sequence that dont converges to some value in any outer space?

Answer 1

If I understood correctly, no. There is a process that can expand any metric space to a complete space, meaning, for every space X there exist a space Y such that X is a subset of Y and Y is complete. To be specific, check out this article https://en.wikipedia.org/wiki/Complete_metric_space under the "completion" header.