48 CHAPTER 2. BASIC TOPOLOGY AND ALGEBRA

Proof: Let{

xk}∞

k=1 be a sequence in Q. Then for each i,{

xki}∞

k=1 is contained in [−r,r] .Therefore, taking a succession of p subsequences, one obtains a subsequence of

{xk}

de-noted as {xnk} such that for each i≤ p,

{xnk

i

}converges to some xi ∈ [−r,r]. It follows that

limk→∞ xnk = x where the ith component of x is xi. ■Since Rp is a metric space, Theorem 2.5.8 implies the following theorem.

Theorem 2.5.11 A nonempty set K contained in Rp is compact if and only if it issequentially compact.

Now the general result called the Heine Borel theorem comes right away.

Theorem 2.5.12 For K ⊆ Rp a nonempty set, the following are equivalent.

1. K is compact.

2. K is sequentially compact.

3. K is closed and bounded.

Proof: The first two are equivalent from Theorem 2.5.5. It remains to show that theseare equivalent to closed and bounded.⇒Suppose the first two hold. Why is K bounded? If not, there is kn ∈ K \B(0,n) .

Then {kn} cannot have a Cauchy subsequence and so no subsequence can converge thanksto Theorem 2.3.3. Why is K closed? Using Corollary 2.2.8, it suffices to show that ifkn→ k, then k ∈ K. We know that {kn} is a Cauchy sequence by Theorem 2.3.2. Since Kis sequentially compact, a subsequence converges to some l ∈ K. However, from Theorem2.3.3, the original sequence also converges to l and so l = k. Thus k ∈ K. The following isanother proof that K is closed given K is compact.⇐Suppose now that K is closed and bounded. Then it is a closed subset of [−r,r]p for

large r. Thus, it is a closed subset of a compact set by Corollary 2.5.10. Therefore, it iscompact by Proposition 2.5.2. ■

As shown above, every closed interval [a,b] is compact and sequentially compact. Nextis an easy observation about the product of compact sets. The proof was essentially usedabove.

Corollary 2.5.13 Suppose Ki is a compact subset of R. Then K ≡∏pi=1 Ki is a compact

subset of Rp.

Proof: This is easiest to see in terms of sequential compactness. Let {xn}∞

n=1 be asequence in K. Say xn =

(x1

n x2n · · · xp

n). By sequential compactness of each Ki, it

follows that taking p subsequences, one can obtain a subsequence, still denoted by {xn}such that for each i≤ p, limn→∞ xi

n = xi ∈ Ki. Then xn→ x ∈ K. ■Since Cp is just R2p, closed and bounded sets are compact in Cp also as a special case

of the above.A useful corollary of this theorem is the following, sometimes called the Weierstrass

Bolzano theorem.

Corollary 2.5.14 Let {xk}∞

k=1 be a bounded sequence in Rp or Cp. Then it has aconvergent subsequence.