1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 103

SUMMARY:

(a) To the calculator, every convergent sequence is eventually con-

stant!

(b) To the calculator, many divergent sequences are eventually con-

stant, h en ce convergent, as far as you can tell from the calculator.

( c) Evidence gathered about convergence by direct calculation may

cause you to conclude, mistakenly, that a divergent sequence con-

verges.

CANTOR'S NESTED INTERVALS THEOREM

We come now to one of t he most famous t heorems of the early twentieth
century in elementary real analysis. It is important because of the pivotal role
it plays in establishing the powerful Bolzano-Weierstrass Theorem in Section
2.6-which in turn plays a key role in establishing deep results about Cauchy
sequences in Section 2.7, the topology of t he real number system in Chapter
3, and continuous functions in Chapter 4 and Section 5.7. It is included here
because it can easily be proved using the monotone convergence theorem.

Theorem 2.5.17 (Cantor's Nested Intervals Theorem) Let {In} be a
sequence of nonempty closed intervals In = [an, bn] such that Ii 2 h 2 · · · 2
00
In 2 · · ·, and lim (bn - an)= 0. Then n In consists of exactly one point.
n-1-00 n=l

In the interest of clarity, we shall state and prove a slightly more gen-
eral version of t his theorem. The reader will see that Cantor's t heorem follows
directly from this alternate version.

Alternate Theorem 2.5.17: Let {In} be a sequence of nonempty closed in-
t ervals In = [an, bn] such that Ii 2 I 2 2 · · · 2 In 2 · · ·. Then

00

(a) n In is a nonempty closed interval, and

n=l

00

(b) if lim (bn - an) = 0, then n In consists of exactly one point.

n-too n=l

Proof. (a) Let {In} be a sequence of nonempty closed intervals In =

[an, bn] that are "nested"; i.e., Ii 2 h 2 · · · 2 In 2 · · ·. Consider the sequences