Convergence of Cauchy's sequence

Question

I understood that every convergent sequence is a Cauchy sequence. It seems that the converse is not necessarily true. An example given is the sequence $\{x_n\}$, where $x_n = (0.1)^n$ is a Cauchy sequence, but not convergent in (0,1). So, Can I safely say that every Cauchy sequence actually converges to some limit and it is said to be convergent only if the limit point is a part of the given metric space? Because the same sequence is convergent on $\mathbb R$, but not in (0,1)

Answer 1

Not always. A space in which every Cauchy sequence is a convergent sequence is called a complete space, but not every imaginable space is complete.

The set of real numbers is complete, which means that a Cauchy sequence of real numbers will have a real limit. Other sets, like interval $(0,1)$ or the set of rational numbers $\mathbb Q$ are not complete, and the Cauchy sequence of numbers from them do not have to have a limit in these sets.

However, when you have a non-complete set, you can always construct its completion, by adding new elements to this set in such a way that the result will be a complete set. So you can say that every Cauchy sequence of elements of some space has a limit in the completion of this set, but not necessarily in the set itself. For example, the completion of interval $(0,1)$ is interval $[0,1]$, and the completion of $\mathbb Q$ is $\mathbb R$.

Answer 2

It is not true that every convergent sequence is a Cauchy sequence by definition. That is not part of the definition of convergent sequence. It is true because it can be proved that it is true.

Given a Cauchy sequence $(x_n)_{n\in\mathbb N}$ in a matric space $(M,d)$, there is a larger space $\left(M^\star,D\right)$ (where $M^\star\supset M$ and $D$ is an extension of $d$) such that, in $\left(M^\star,D\right)$, the sequence $(x_n)_{n\in\mathbb N}$ converges. It is not hard to prove, but it is not a trivial statement.

Answer 3

The condition that every Cauchy sequence in $X$ converges to a point in $X$ is the definition of complete metric space.

A closed subset of a complete space is a complete space in its subspace topology.

Answer 4

Whether or not a sequence of points in a metric space $M$ is Cauchy depends only on the metric.

If it converges to a point in $M$ then it is convergent (of course). Conversely, every sequence that converges to a point in $M$ is Cauchy.

But it doesn't make sense in general to refer to a limit that's not in $M$. The open unit interval is naturally a subset of the reals, but abstract metric spaces $M$ don't come as subsets of larger spaces that contain the limits of Cauchy sequences that don't converge in $M$. You can always build such spaces, but that's not what you are asking about.