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This article is NOT about Cantor's diagonal argument! |
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:''b''<sub>1</sub> = ''x''<sub>''i''</sub>, where ''i'' is the smallest index such that ''x''<sub>''i''</sub> is not equal to ''a''<sub>1</sub>.
:''a''<sub>''n''+1</sub> = ''x''<sub>''i''</sub>, where ''i'' is the smallest index ''greater than the one considered in the previous step'' such that ''x''<sub>''i''</sub> is between ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub>.
:''b''<sub>''n''+1</sub> = ''x''<sub>''i''</sub>, where ''i'' is the smallest index ''greater than the one considered in the previous step'' such that ''x''<sub>''i''</sub> is between ''a''<sub>''n''+1</sub> and ''b''<sub>''n''</sub>.
The two monotone sequences ''a'' and ''b'' move toward each other. By the "gaplessness'' of '''R''', some point ''c'' must lie between them. The claim is that ''c'' cannot be in the range of the sequence ''x'', and that is the contradiction. If ''c'' were in the range, then we would have ''c'' = ''x''<sub>''i''</sub> for some index ''i''. But then, when that index was reached in the process of defining ''a'' and ''b'', then ''c'' would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.
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