In sci.math, chajadan@mail.com <chajadan@mail.com> wrote on 31 May 2007 00:19:45 -0700 <1180595985.632003.291450@d30g2000prg.googlegroups.com>: > >> >> O.k., a=0.9999... is a real number. b=1 is a real number. >> >> What value has b-a? > >> >> Karl- > > > I do not know that 0.999... is a real number. > > --charlie >
I don't see why it couldn't be.
Dedekind: Take two sets. A is a union of the following sets, evaluated under Q (not R):
A_0 = {x: x < 0} A_1 = {x: x < 0.9} A_2 = {x: x < 0.99} ...
B is the set of all rational numbers equal to or exceeding 1.
Clearly, A contains no greatest element, A is closed downwards, B is closed upwards. Between them, A union B should be Q. Therefore, we have a cut, and 0.999... is the real number defined by that cut on the lower side.
Cauchy Sequences.
Clearly, 0.999... is the limit of the following sequence.
0.9 = 9/10 0.99 = 99/100 0.999 = 999/1000 ...
Therefore, 0.999... is a real number.
Whether 0.999... = 1 I've not yet established here, of course.
-- #191, ewill3@earthlink.net Q: "Why is my computer doing that?" A: "Don't do that and you'll be fine."
