What sequence would be an example of a Cauchy sequence that diverges?

Question

In metric space every convergent sequence is a Cauchy sequence, but what about the converse?

Take $X=(0,1)$ with the usual metrics and $x_n=\frac1n$. (This was asked several times on the site.) — Did, Jan 12 '17 at 18:02

score 2 · Accepted Answer · answered Jan 12 '17 at 18:04

Consider the metric space $\mathbb R\setminus\{0\}$ with the usual distance, and the sequence $$ 1, \frac12, \frac13, \frac14, \frac15, \ldots $$

In $\mathbb R$ this converges to $0$ (and so it is Cauchy), but $0$ is not an element of the metric space, so it is not convergent as a sequence in $\mathbb R\setminus\{0\}$.

score 0 · Answer 2 · answered Jan 12 '17 at 18:25

0

The sequence $(x_n)$ defined by

$$x_{n+1}=\left(x_n+\frac{2}{x_n}\right)\frac12$$

for $x_0>0$ converge to $\sqrt 2$. But if our metric space is $\Bbb Q$ then the sequence doesnt converge on this space.

answered Jan 12 '17 at 18:25

What sequence would be an example of a Cauchy sequence that diverges?

2 Answers2