According to Wikipedia:
A metric space $M$ is said to be complete if every Cauchy sequence
converges in $M$
$ $
A metric space $M$ is compact if every sequence in $M$ has a
subsequence that converges to a point in $M$
I can't seem to find a situation where a complete metric space is not compact or vice versa.
First of all, why can't we say $M$ is complete if every sequence converges in $M$. Since if a sequence converges to a point outside $M$ it is clearly not complete?
And so if every sequence converges in $M$, then clearly every sequence has a subsequence which converges in $M$ and hence it is also compact (if it is complete).
And if every sequence has a subsequence which converges to $M$ then doesn't the sequence itself converge to $M$ in which case the compact space is also complete?
Apologies if I'm totally off track.
If someone could provide me with some examples which show the difference between the two I'd be very grateful. Preferably an example that gives me a much better intuitive understanding, because I think my main problem is intuition, I go a bit mad trying to understand rigorous definitions.
Best Answer
The real line $\mathbb{R}$ is complete but not compact. The key word in the definition of completeness is "Cauchy". Note that the definition of compactness does not speak of Cauchy sequences but rather of arbitrary sequences.
Here $\mathbb{R}$ is not compact because the sequence $u_n=n$ does not contain a convergent subsequence.