Interpretation of non-convergence in metric spaces

Question

1 .From analysis I know a sequence can either have limit or not. If it has a limit it is said convergent if the limit is finite, divergent if it is $+\infty $ or $-\infty$. Therefore if someone tells me a sequence is not convergent, I can't help but think that it can either be divergent or its limit does not exist.

Why then in the context of Cauchy sequences, people seem to interpret "$(x_n)$ does not converge as " it converges but outside the space " instead of the previous interpretation?

I know that "In a metric space $(X,d)$, every convergent sequence is a Cauchy sequence"

So if a sequence converges outside the space, why people/textbooks say "it doesn't converge" when in fact it does (outside the space, but it does!) and leave out the other more natural possibilities ( divergence or of inexistent limit)? If they always wrote "it doesn't converge in the space X", then I wouldn't argue.

Answer 1

I don't doubt that you have described correctly the concpet of convergent (and divergent) sequence as you have learned it, but you should keep in mind that, by far, the usual definition is: a sequence (of real numbers) converges if it has a limit and it diverges otherwise. So, for instance, the sequence $\bigl((-1)^n\bigr)_{n\in\Bbb N}$ diverges.

Now, if you are working on a metric space $(X,d)$, you can say that a certain sequence $(x_n)_{n\in\Bbb N}$ doesn't converge on $X$, but that's redundant, since, if you are working on $(X,d)$, then $X$ is your whole universe. So, you can simply say that it doesn't converge (or that it diverges).

It turns out that for every metric space $(X,d)$ and for every Cauchy sequence of elements of $X$, there is a larger space $\left(\overline X,D\right)$ such that, on that space, the sequence converges. Note that asserting that $\left(\overline X,D\right)$ is a larger space means two things:

• $X\subset\overline X$;
• if $x_1,x_2\in X$, then $D(x_1,x_2)=d(x_1,x_2)$.

In particular, the sequence $(x_n)_{n\in\Bbb N}$ may well converge on $\left(\overline X,D\right)$ and diverge on $(X,d)$. The concept of convergence depends upon the space that you are working with.