Meaning of being a Cauchy sequence

Question

A Cauchy sequence in a metric space $(X,d)$ is a sequence for which the distance between two terms can be made as small as we want, provided we look far enough in the sequence.

Let $X \subseteq Y$, where $Y$ is a set on which $d$ can be extended. Is it always true that for any Cauchy sequence $\{x_n\}$ in $X$ one can find $y\in Y$ such that $d(x_n,y)\rightarrow 0$?

In other words, is it true that every Cauchy sequence "converges" to something? (and if this something is in $X$ we say it is convergent?)

Answer 1

I think what you're looking for is the concept of the completion of the metric space $X$.

Here you construct $Y$ by adding to $X$ an "artificial" limit element for each Cauchy sequence that doesn't already have one in $X$ -- or rather, for each equivalence class of such Cauchy sequence where sequences $(x_n)$ and $(y_n)$ are related if $d(x_n,y_n)\to 0$.

One main example of this is that $\mathbb R$ is a metric completion of $\mathbb Q$.

Answer 2

No, in order to show that a Cauchy Sequence converges, a more delicate approach would be needed and that's not always the case. If, though, one proves that for a space $(X,d) \subseteq (Y,d)$ it is $d(x_n,y) \to 0$ for a Cauchy Sequence $\{x_n\}_{n \in \mathbb N} \in X$, that means that the space $X$ equipped with the metric $d$ is complete, thus Banach (if the metric $d$ is a norm of course, thus we are dealing with a complete normed space).