How to show a metric space is not complete
It can be used when you are aware of the existence of a metric space $(Y,d^\ast)$ such that:
- $X\subset Y$;
- $(\forall x,x'\in X):d^\ast(x,x')=d(x,x')$.
In the example that you have mentioned, that space is $\mathbb R$, endowed with its usual metric.
Actually, such a metric space always exist (take the completion of $X$, for instance), but if you don't know how to work with it, that's a useless piece of information.
The point here is that there are two metric spaces involved.
Basically, the following situation happens. Let $X$ be a metric space and $Y$ be a metric subspace of $X$ (thus $Y \subseteq X$ and the distance in $Y$ is the distance in $X$ restricted to $Y$).
In our situation we have a sequence $(y_n)_n$ in $Y$ and this converges to a point $x \notin Y$. But thus $(y_n)_n$ is a convergent sequence in $X$ and thus Cauchy in $X$ and thus Cauchy in $Y$. However, if $(y_n)_n$ would converge in $Y$, there is a limit $y \in Y$ and by uniqueness of limits we get $y =x \notin Y$, which is impossible.
Thus, such a sequence cannot converge in $Y$ and we have found a Cauchy sequence that does not converge. Hence, $Y$ can't be complete.