Let me be more precise about what I mean by equivalence here: in the presence of the ordered field axioms, the least upper bound property is equivalent to the property that every infinite decimal corresponds to an element of the field (here "corresponding" means "belonging to all the 10-adic intervals that are tacit in the decimal expansion"), and any ordered field satisfying either of these "completeness properties" is isomorphic to the real numbers. (Technical caveat: in addition to the ordered field axioms, you also need some axioms about the natural numbers, since the natural numbers are used to index the successive digits.). See jamespropp.org/reverse.pdf for details. Jim On Wednesday, January 30, 2013, Dan Asimov wrote:
"The two points of view can be shown to be equivalent" is very hard to believe, without a great deal more detail about just what the "address on the number line" concept means.
--Dan
<< On the subject of what "0.999..." _means_ (which we need to agree on before we can discuss what number it denotes), let me suggest that instead of thinking of it as the sum of an infinite series, one can think of it as an address on the number line. The two points of view can be shown to be equivalent, but pedagogically they're different.
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