Cantor's first set theory article: Difference between revisions

Revision as of 15:59, 20 February 2007

Revision as of 16:05, 20 February 2007

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:Let ''b''_''n''+1 be the first element in the sequence ''x'' which is strictly between ''a''_''n''+1 and ''b''_''n''.

The two [[monotonic function|monotone sequences]] ''a'' and ''b'' move toward each other. By the completeness of '''R''', some point ''c'' must lie between them. (Define ''A'' to be the set of all elements in '''R''' that are smaller than some member of the sequence ''a'', and let ''B'' be the [[complement]] of ''A''; then every member of ''A'' is smaller than every member of ''B'', and so property 4 yields the point ''c''.) Since ''c'' is an element of '''R''' and the sequence ''x'' exhaustsrepresents all of '''R''', we must have ''c'' = ''x''_''i'' for some index ''i''; there must exist an ''x''_''i'' in the ''x'' sequence, corresponding to ''c''. But, when that index was reached in the process of defining the sequences ''a'' and ''b'', then ''c'' would have been added as the next member of one or the other sequence, contrary to the fact that ''c'' lies strictly between the two sequences. This contradiction finishes the proof.

==Real algebraic numbers and real transcendental numbers==