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Blast into Math! Analatic nummer theora: ants, ghosts and giants

Theorem 7.4.1 (Decimal Expansion Theorem) Let x be a real number with 0 ≤ x< 1. Then there exists

∞ with each x n ∈{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} such that
a unique sequence {x n } n=1
∞
x n
x = ,
10 n
n=1
and such that there is no N ∈ N with

x n =9 ∀n ≥ N.

Proof: Let’s see how for each real number x between 0 and 1 we can choose the digits of its decimal
expansion. To do this, we can use a set. Since x ≥ 0, the set

s

S = s ∈ Z such that 0 ≤ s ≤ 9, and ≤ x = ∅,
10

because 0 ∈ S . The set S is a set of integers which by its definition is bounded above by 9. Therefore
S has a unique largest element. We define the first digit in the decimal expansion of x to be this unique
largest element of S and call this digit x 1 . Then since x 1 ∈ S ,

x 1
≤ x.
10

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