Theorem: The set of real numbers between 0 and 1, (0,1)={𝑥∈ℝ | 0<𝑥<1} is uncountable.


1. Suppose (0,1) is countable.
2. Since it is not finite, it is countably infinite.
3. We list the elements 𝑥_𝑖 of (0,1) in a sequence as follows:
𝑥_1=0. 𝑎_11 𝑎_12 𝑎_13⋯𝑎_1𝑛 ⋯
𝑥_2=0. 𝑎_21 𝑎_22 𝑎_23⋯𝑎_2𝑛 ⋯
𝑥_3=0. 𝑎_31 𝑎_32 𝑎_33⋯𝑎_3𝑛 ⋯
⋮
𝑥_𝑛=0. 𝑎_𝑛1 𝑎_𝑛2 𝑎_𝑛3⋯𝑎_𝑛𝑛 ⋯
⋮
where each 𝑎_𝑖𝑗∈{0,1,⋯,9} is a digit.*
4. Now, construct a number 𝑑=0. 𝑑_1 𝑑_2 𝑑_3⋯𝑑_𝑛 ⋯ s.t.
𝑑_𝑛={(1, if 𝑎_𝑛𝑛≠1, 2 if 𝑎_𝑛𝑛=1.)}
5. Note that ∀𝑛∈ℤ^+,𝑑_𝑛≠𝑎_𝑛𝑛. Thus, 𝑑 ≠ 𝑥_𝑛,∀𝑛∈ℤ^+.
6. But clearly, 𝑑∈(0,1), hence a contradiction. Therefore (0,1) is uncountable.

Illustration:
0.20148802… 𝑑_1 is 1 because 𝑎_11=2
0.11666021… 𝑑_2 is 2 because 𝑎_22=1
0.03853320… 𝑑_3 is 1 because 𝑎_33=8
0.96776809… 𝑑_4 is 1 because 𝑎_44=7
0.00031002… 𝑑_5 is 2 because 𝑎_55=1
Hence 𝑑=0.12112…, which is not in the list. So, the list is incomplete. This is true regardless of how the elements in (0,1) are listed