If m s then there exists anx s such thatm x m which

(Heine-Borel Theorem) A subset S of R is compact if and only if it is closed
and bounded.
Proof: Suppose S is compact. Let G be the collection of open intervals I
n
= ( - n, n ) , n = 1 , 2 , . . . .
Then G is an open cover of S . Since S is compact, G contains a finite subcover I
n
1
, I
n
2
, . . ., I
n
k
.
Let m = max n
i
. Then S ⊆ I
m
, and for all x ∈ S, | x | ≤ m . Therefore S is bounded.
To show that S is closed, we must show that S contains all its accumulation points. Suppose
that p is an accumulation point of S and suppose p / ∈ S . For each positive integer n , let
G
n
= [ p - 1 /n, p + 1 /n ]
c
. Since the complement of a closed interval is an open set, G
n
is an open
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(Bolzano-Weierstrass Theorem) If S ⊆ R is a bounded infinite set, then S
has at least one accumulation point.
Proof: Let S be a bounded infinite set and suppose that S has no accumulation points. Then S
is closed (vacuously), and S is compact. For each x ∈ S , let N
x
be a neighborhood of x such that
S ∩ N
x
= { x } . The set of neighborhoods N
x
, x ∈ S is an open cover of S . Since S is compact,
this open cover has a finite subcover N
x
1
, N
x
2
, . . ., N
x
k
. But S ∩ [ N
x
1
∪ N
x
2
∪ . . . ∪ N
x
k
] =
{ x
1
, x
2
, . . ., x
k
} which implies that S is finite, a contradiction.
Exercises 1.5
1. True – False. Justify your answer by citing a theorem, giving a proof, or giving a counter-
example.
(a) Every finite set is compact.
(b) No infinite set is compact.
(c) If a set is compact, then it has a maximum and a minimum.
(d) If a set has a maximum and a minimum, then it is compact.
(e) If S ⊆ R is compact, then there is at least one point x ∈ R such that x is an
accumulation point of S .
(f) If S ⊆ R is compact and x is an accumulation point of S , then x ∈ S .
2. Show that each of the following subsets S of R is not compact by giving an open cover of
S that has no finite subcover.
(a) S = [0 , 1)
(b) S = N
(c) S = { 1 /n : n ∈ N }
3. Prove that the intersection of any collection of compact sets is compact.
4. Prove that if S ⊆ R is compact and T is a closed subset of S , then T is compact.
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