# Interactive Real Analysis

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## 2.2. Uncountable Infinity

### Proposition 2.2.1: An Uncountable Set

The open interval (0, 1) is uncountable.

### 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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