Page 193 - 'Blast_Into_Math

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

We can keep doing this by defining

x n =1 if the n th elementof s n is 0,

x n =0 if the n th elementof s n is 1.

Then, as it is defined, this sequence is not in S, because it is different from s n forall n ∈ N. But, the set
notin
S contains all sequences of 0 and 1, so we have a contradiction. Consequently, S cannot be countable.

♥

Based on this proposition and geometric series, we can prove that there are uncountably many real
numbers.

Theorem 7.4.3 (Uncountably Many Real Numbers). The set of all real numbers is uncountable.

Proof: For each sequence

∞
{x n } n=1 ∈ S,

since x n =0 or 1 for each n , for each N ∈ N , the partial sum

N N
x n
1
S N = ≤ T N = .
10 n 10 n
n=1 n=1
Since

1/10 − (1/10) N+1 1
T N = < ,
1 − 1/10 9

1
T N < , ∀N ∈ N.
9

Since S N ≤ T N ,

1
S N < ∀N ∈ N.
9

Each term

x n
10 n

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