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===The proof===
The proof is by contradiction. It begins by assuming '''R''' is countable and thus that some [[sequence]] ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... has all of '''R''' as its range. Define two other sequences (''a''<sub>''n''</sub>) and (''b''<sub>''n''</sub>) as follows:
:Pick ''a''<sub>1</sub> < ''b''<sub>1</sub> in '''R''' (possible because of property 2).
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