# Cauchy sequences that do not converge?

• Mar 20th 2011, 07:39 PM
tonybruguier
Cauchy sequences that do not converge?
Hello,

I was under the impression that Cauchy sequences do not always converge. That is NOT true in $\displaystyle \mathbb{R}$ where every Cauchy sequence converges.

I was under the impression that there were some spaces where this is NOT true. There are some spaces where Cauchy sequences do NOT have a limit. However, I cannot find an example on the Wikipedia page:
https://secure.wikimedia.org/wikiped...auchy_sequence

I am asking this, because I have been reading this PDF document:

If you look at pages 20 and 21, you will see that at the very top of page 21, it seems to indicate that every sequence that is Cauchy has a limit.

So what is the correct statement? Are all Cauchy sequences convergent? If yes, is the proof in the PDF a good proof? If not, do you have a counter example and why the derivation of the PDF do not work?

I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in $\displaystyle \mathbb{Q}$ that have a limit in $\displaystyle \mathbb{R}$.

• Mar 20th 2011, 11:31 PM
Drexel28
Quote:

Originally Posted by tonybruguier
Hello,

I was under the impression that Cauchy sequences do not always converge. That is NOT true in $\displaystyle \mathbb{R}$ where every Cauchy sequence converges.

I was under the impression that there were some spaces where this is NOT true. There are some spaces where Cauchy sequences do NOT have a limit. However, I cannot find an example on the Wikipedia page:
https://secure.wikimedia.org/wikiped...auchy_sequence

I am asking this, because I have been reading this PDF document:
I currently suspect that it has something to do with density. Like, we can find Cauchy sequences in $\displaystyle \mathbb{Q}$ that have a limit in $\displaystyle \mathbb{R}$.
This is a property of whatever metric space you're working in. A metric space is called complete if it has the property that every Cauchy sequence converges. For example, $\displaystyle \mathbb{R}^n$ and $\displaystyle \mathbb{C}^n\approx\mathbb{R}^{2n}$ are complete for each $\displaystyle n\in\mathbb{N}$. A less trivial example is $\displaystyle C([a,b])$--the space of all continuous functions equipped with the metric induced by $\displaystyle \|\cdot\|_{\infty}$. That said, as you indicated $\displaystyle \mathbb{Q}$ is not complete. There are hunderds of examples I'm sure you could think of (e.g. $\displaystyle \left(1+\frac{1}{n}\right)^n$, $\displaystyle \displaystyle \sum_{m=1}^{n}\frac{1}{m^2}$, etc.)