```Date: Dec 2, 2013 8:34 AM
Author: G. A. Edgar
Subject: Re: Uncountability of the Real Numbers Without Decimals

In article <38ef6b5b-8df9-430a-9e5d-2867d4fde621@googlegroups.com>,Zeit Geist <tucsondrew@me.com> wrote:> The following is a Proof of the Uncountability of the Set of real Numbers. > Please note that it avoids the use the Representation of the Real Numbers as> Infinite Decimals.  It uses only the Property of Completeness (that every Set> of Real Numbers that is bounded above has a Least Upper Bound which is a Real> Number); and some Properties due to the Ordering of the Real Numbers, such as> if x is Real Number then there exist Real Numbers a and b such that a < b and> x ~e [ a, b ].> > The proof proceeds by Contradiction.  We assume the Set of Real Numbers is> Countable, and thus can be exhausted in a Sequence.> > Take any such Sequence of Real Numbers, X = { x_n | n e N }.  We begin by> choosing Real Numbers, a_1 and b_1, such that a_1 < b_1 and x_1 ~e [ a_1, b_1> ].  Next, we choose Real Numbers, a_2 and b_2, such that a_1 < a_2 < b_2 <> b_1 and x_2 ~e [ a_2, b_2 ].  We continue by choosing Real Numbers a_k and> b_k for every k e N, such that for every k e N, we have > a_k-1 < a_k < b_k < b_k-1 and, of course, x_k ~e [ a_k, b_k ].> > Doing so for every Natural Number, we define A = { a_n | n e N }.  Now, A is> a Set of Real Numbers that is bounded above, since any b_n is an upper bound> of A.  Hence, a = sup(A) is a Real Number.  Since our Sequence, X, exhausts> all Real Numbers, a e X and there is a Natural Number, m, such that x_m = a.> > Now, we have previously defined Real Numbers, a_m and b_m, such that a = x_m> ~e [ a_m, b_m ].  However, we know a_m <= a, since a = sup(A) and a e A; and> a <= b_m, since any b_n is an upper bound of A.  These together give us that> we must have x_m e [ a_m, b_m ].  This results in a Contradiction.  Hence, we> must have that the Set of Real Numbers is Uncountable.> > qed> > I find the proof rather straight forward.  Question, comments, suggestions> and corrections are welcome.> > ZGSimilar to Cantor's first proof of uncountability of the real line.-- G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/
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