Krdytkyu

I need to prove that every compact metric space is complete. I think I need to use the following two facts:

1. A set $K$ is compact if and only if every collection $mathcal{F}$ of closed subsets with finite intersection property has $bigcap{F:Finmathcal{F}}neqemptyset$.


2. A metric space $(X,d)$ is complete if and only if for any sequence ${F_n}$ of non-empty closed sets with $F_1supset F_2supsetcdots$ and $text{diam}~F_nrightarrow0$, $bigcap_{n=1}^{infty}F_n$ contains a single point.

I do not know how to arrive at my result that every compact metric space is complete. Any help?

Thanks in advance.

Let $langle F_nrangle_{ninBbb{N}}$ be a descending sequence of nonempty closed sets satisfy that $operatorname{diam} F_nto 0$ as $ntoinfty$. You can easily check that if $m_1<m_2<cdots<m_k$ then
$$
F_{m_1}cap F_{m_2}capcdotscap F_{m_k} =F_{m_k}neq varnothing
$$
so $langle F_nrangle_{ninBbb{N}}$ satisfies finite intersection property. Since $(X,d)$ is a compact metric space, $bigcap_{ninBbb{N}} F_n$ is not empty. Since
$$
operatorname{diam} bigcap_{ninBbb{N}} F_n le operatorname{diam} F_mto 0qquad text{as }> mtoinfty
$$
so $bigcap_{ninBbb{N}} F_n$ contains at most one point. So $bigcap_{ninBbb{N}} F_n$ is singleton.

If you do not wish to use the Heine-Borel theorem for metric spaces (as suggested in the answer by Igor Rivin) then here is another way of proving that a compact metric space is complete:

Note that in metric spaces the notions of compactness and sequential compactness coincide. Let $x_n$ be a Cauchy sequence in the metric space $X$. Since $X$ is sequentially compact there is a convergent subsequence $x_{n_k}to x in X$.

All that now remains to be shown is that $x_n to x$. Since $x_{n_k}to x$ there is $N_1$ with $n_k ge N_1$ implies $|x_{n_k}-x|<{varepsilonover 2}$. Let $N_2$ be such that $n,mge N_2$ implies $|x_n-x_m|<{varepsilon over 2}$.

Then $n>N=max(N_1,N_2)$ implies
$$ |x_n-x|le |x_n-x_N|+|x_N-x|<varepsilon$$

Hence $X$ is complete.

Here's a not-so-fancy way:

Let ${a_n}$ be a Cauchy sequence.

If the set of values in the (image of the) sequence is finite, then use the Cauchy criterion to show that the sequence is eventually constant (and so converges).

If the set of values of the sequence is infinite, then use compactness to finite a limit point of this set. Use this limit point to construct a convergent subsequence of the original sequence. Then use the Cauchy criterion to show the original sequence converges to the same limit as the subsequence.

This follows from Heine-Borel (see the wiki page for the relevant proofs).

Let $X$ be a compact metric space and let ${p_n}$ be a Cauchy sequence in $X$. Then define $E_N$ as ${p_N, p_{N+1}, p_{N+2}, ldots}$. Let $overline{E_N}$ be the closure of $E_N$. Since it is a closed subset of compact metric space, it is compact as well.

By definition of Cauchy sequence, we have $lim_{Ntoinfty} text{diam } E_N = lim_{Ntoinfty} text{diam } overline{E_N} = 0$. Let $E = cap_{n=1}^infty overline{E_n}$. Because $E_N supset E_{N+1}$ and $overline{E_N} supset overline{E_{N+1}}$ for all $N$, we have that $E$ is not empty. $E$ cannot have more than $1$ point because otherwise, $lim_{Ntoinfty} text{diam } overline{E_N} > 0$, which is a contradiction. Therefore $E$ contains exactly one point $p in overline{E_N}$ for all $N$. Therefore $p in X$.

For all $epsilon > 0$, there exists an $N$ such that $text{diam } overline{E_n} < epsilon$ for all $n > N$. Thus, $d(p,q) < epsilon$ for all $q in overline{E_n}$. Since $E_n subset overline{E_n}$, we have $d(p,q) < epsilon$ for all $q in E_n$. Therefore, ${p_n}$ converges to $p in X$.

Hint: If you have a sequence wherein $d(a_{N},a_{k})<r$ for all $kgeq N$, then the limit of that sequence, if exist, must be in the closure of the open ball radius $r$ around $a_{N}$.