1549901369-Elements_of_Real_Analysis__Denlinger_

3.4 *The Cantor Set 167

(d) 0.1111111 · · · (bas; 3) = i + i + 217 + · · ·

• 1 1
  ~ 3 ~ ~~ - ·
  1-i 3-1 2'

(

Recall t: geometri~ serie.s from calculus:)

L arn = 1 _ r if Ir! < 1

n=O

(e) 0.020202020 · · · (base 3) = ~ + ffe + ffe + · · ·

_

00

2 (l)n

-2::.::- -

n=O 9 9

2

9

1-i

2 .1

9-1 4

D

Theorem 3.4.10 The Cantor set consists of all those real numbers in [O, 1]

that can be represented by a base-3 decimal consisting of only 0 's and 2 's.

Proof. Exercise 5. •

Theorem 3.4.10 allows us to show that the Cantor set contains many real

numbers besides endpoints of intervals removed in the construction.

Example 3.4.11 i is in the Cantor set, but is not an endpoint of a "removed

interval."

Proof. From Example 3.4.9, i = 0.202020 · · · (base-3), so by Theorem

3.4.10, i E C. However, i is not an endpoint of a "removed interval,'' since

any such endpoint is a fraction whose denominator is a power of 3. D

Theorem 3.4.10 allows us to prove other interesting facts about the Cantor

set.

PROPERTIES OF THE CANTOR SET

We have already seen that the Cantor set is compact.

Theorem 3.4.12 The Cantor set has the same number of elements as [O, 1];

hence, it is uncountable.

Proof. Define the function f : C ----t [O, 1] as follows. Let x E C. By
Theorem 3.4. 10 , x has a ternary decimal representation consisting of all O's
and 2's. Replace all 2's by l's. The result can be regarded as a base-2 decimal