How can a sequence be Cauchy and divergent at the same time?

Question

The sequence $(x_n) = 1/n$ is not convergent in $(\mathbb{R} \backslash \{0\}, | · |)$ (the usual distance).

Knowing that a sequence is Cauchy if there exists an N such that n,m>N we have |x(n)-x(m)| < ε.

Is the reason the sequence is divergent in $(\mathbb{R} \backslash \{0\}, | · |)$ is because the limit point $(x=0)$ is not in the set? As in it is not closed (by sequence characterization)?

Thank you!

Answer 1

The sequence $\left(\frac1n\right)_{n\in\Bbb N}$ is divergent in $\Bbb R\setminus\{0\}$ because there is no $x\in\Bbb R\setminus\{0\}$ such that $\lim_{n\to\infty}\frac1n=x$. The fact that $\Bbb R\setminus\{0\}$ is not a closed subset of $\Bbb R$ is relevant, but not essential. After all $(-2,2)$ is also a subset of $\Bbb R$ whay is not closed, but the sequence $\left(\frac1n\right)_{n\in\Bbb N}$ converges in it.

Answer 2

This is correct.

A space where all Cauchy sequences converge to a point in the space is called complete. If you have a space that is not complete, such as $\mathbb{R}\backslash\{0\}$, the Cauchy sequences in this space might not converge to a sequence in the space, as you've found. However, we can find a completion of these spaces, in this case all of $\mathbb{R}$, and Cauchy sequences will converge to an element in this space.

In other words, all Cauchy sequences converge to something, and the question is whether or not the limit is in the space itself.

Answer 3

It doesn't converge to a value because the point that it should converge to was removed from the set we are working with.

This is why we call metric spaces complete when every cauchy sequence is convergent. Because when we have a cauchy sequence that doesn't converge we can imagine that there is a point missing (the point to which that sequence should converge).

Answer 4

You've made an interesting observation! This notion is something that is often considered in analysis-- a metric space (e.g. the real numbers $\mathbb{R}$ equipped with the metric $|\cdot|$) is said to be ${\bf complete}$ if each Cauchy sequence has a limit in the space. $\mathbb{R} / \{ 0\}$ is not complete. $\mathbb{R}$ is defined in such a way that it is complete with the metric $| \cdot |$. A standard definition of $\mathbb{R}$ is that it is the completion of the rational numbers based on the metric $| \cdot |$.