I was under the impression that Cauchy sequences do not always converge. That is NOT true in 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:
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 that have a limit in .
Thanks in advance,