Example of a complete metric space which is not compact Is there any example of a complete metric space which is not compact? Why? 
 A: $\mathbb{R}$ is complete in its standard metric, but not compact. The open cover
$$\mathbb{R}=\cdots\cup(-3,-1)\cup(-2,0)\cup (-1,1)\cup (0,2)\cup (1,3)\cup\cdots$$
has no finite subcover. For a different argument: the sequence $1,2,3,\ldots$ has no convergent subsequence (and for metric spaces, compact $\iff$ sequentially compact). 
A: in a metric space you may define Cauchy sequences and convergent sequences in the usual way.
$\{a_i\}$ is Cauchy iff $\forall \epsilon \gt 0 \exists N.m,n \gt N \Rightarrow d(a_m,a_n) \lt \epsilon $
$\{a_i\}$ converges to $a$ iff $\forall \epsilon \gt 0 \exists N.m \gt N \Rightarrow d(a_m,a) \lt \epsilon$ 
(sequential) compactness requires that every sequence contains a convergent subsequence
completeness requires that every Cauchy sequence is convergent.
since compactness implies, in particular, that a Cauchy sequence contains a convergent subsequence, then (in the metric space context) a compact set must be complete. the reverse need not be the case - completeness is a statement about Cauchy sequences, whereas compactness is a statement about all sequences. 
the standard counterexamples are unbounded sequences in complete spaces.
