The Wikipedia page on complete metric spaces gives various examples of metric spaces that are and are not complete – http://en.wikipedia.org/wiki/Complete_metric_space
Here's a few lines in particular –
The open interval $(0, 1)$, again with the absolute value metric, is not
complete either. The sequence defined by $x_n = \frac{1}{n}$ is Cauchy, but does
not have a limit in the given space. However the closed interval $[0,
1]$ is complete; the given sequence does have a limit in this interval
and the limit is zero.
I know that a metric space M is complete if every Cauchy sequence of points in M has a limit that is also in M, but that example above just considers one Cauchy sequence and then announces that the interval is complete.
How can they say it is complete without considering all possible Cauchy sequences in the interval which is what the definition demands..and for that matter, how would it be possible to consider all Cauchy sequences in an interval given that, I presume, there are an infinite number of them?
Can anyone clear this up for me…I have a feeling I'm overlooking something straightforward.
Best Answer
To show that a closed interval is complete, you can do the following: You first show that every Cauchy-sequence is bounded. It therefore has a convergent subsequence and the limit is in the interval since it is closed. Then you verify that if a sub-sequence of a Cauchy sequence converges to a point, the sequence itself converges to the same point.