Why Read This Book? - Index of

5.4 Interior, Exterior, Boundary, and Cluster Points 177
EXERCISE 5.4.5 Show that zero is a cluster point of the following sets.
(a) {1/n} ∞ n=1
(b) [0, 1]
The definition of cluster point motivates a new term for convenience.
Definition 5.4.6 For a real number x and ɛ>0, the set Nɛ(x) −{x} is called
the deleted ɛ-neighborhood of x and is denoted DNɛ(x). Another way to write
DNɛ(x) is {y ∈ R : 0 < |y − x| 0, A ∩ DNɛ(x) is nonempty.
Cluster points have other common names, for example, limit points or
accumulation points.
EXERCISE 5.4.7 Suppose L is the LUB of a set A and L/∈ A. Then L is a cluster
point of A.
EXERCISE 5.4.8 What does it mean for x not to be a cluster point of A?
If x ∈ A, but x is not a cluster point of A, we say that x is an isolated point of A.
Notice in the proof of the next theorem how we must replace an arbitrarily chosen
ɛ>0 with ɛ2 > 0, which might be smaller. The desired point that we find in
the ɛ2-neighborhood will therefore also be in the ɛ-neighborhood.
Theorem 5.4.9 If A is an open set, then every element of A is a cluster point.
Proof. Suppose A is open and pick any x ∈ A. To show x is a cluster point of A,we
must first choose ɛ>0. Since A is open, there exists ɛ1 > 0 such that Nɛ1 (x) ⊆ A.
Let ɛ2 = min{ɛ, ɛ1}. Then the point x + ɛ2/2 ∈ A ∩ DNɛ(x).
EXERCISE 5.4.10 Show that the converse of Theorem 5.4.9 is not true.
Sometimes one statement might seem stronger than another but is actually
equivalent. In the next exercise, the fact that every deleted neighborhood contains
some point of A is actually strong enough to allow you to show that every deleted
neighborhood contains infinitely many points of A.
EXERCISE 5.4.11 If x is a cluster point of A, then every ɛ-neighborhood of x
contains infinitely many elements of A. 16
16 Try a proof by contrapositive.