MathCS.org - Real Analysis: Proposition 2.2.1: An Uncountable Set

Proposition 2.2.1: An Uncountable Set

The open interval (0, 1) is uncountable.

Context

Proof:

Any number x in the interval (0, 1) can be expressed as a unique, never-ending decimal. Actually, this is not quite true: 0.1499999... is the same number as 0.15000.... But when we simply discard those numbers with a non-ending tail of 9's we still get the open interval (0, 1), and now every number has a unique decimal representation. If these numbers were countable, we could list them in a two-way infinite table:

1. number: x₁¹, x₂¹, x₃¹, x₄¹, ...
2. number: x₁², x₂², x₃², x₄², ...
3. number: x₁³, x₂³, x₃³, x₄³, ...
4. number: x₁⁴, x₂⁴, x₃⁴, x₄⁴, ...
...

where each expression in parenthesis represents all decimals in the decimal representation of a particular number without the leading '0.'.

In this list, what would be the number associated to the following element:

Let x be the number represented by (x₁, x₂, x₃, x₄, ...), where we let:
- x₁ = 0 if x₁¹ = 1 and x₁ = 1 if x₁¹ = 0
- x₂ = 0 if x₂² = 1 and x₂ = 1 if x₂² = 0
- x₃ = 0 if x₃³ = 1 and x₃ = 1 if x₃³ = 0
- ...

This new element x is different from the first one in our list, because they differ in their first entry; x is different from the second one in the list, because they differ in the second entry; x is different from the third one because they differ in the third entry, etc. But now it is clear that x can not be in the above list, because it differs with the n-th element of that list in the n-th entry. But this element represents a number in the interval (0, 1). Hence, we have found that we were unable to list all numbers in (0,1), and therefore the interval is indeed uncountable.

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Proposition 2.2.1: An Uncountable Set

Proof: