In standard mathematics there is no such thing as adjacent real numbers. Endless arguments from non-mathematicians notwithstanding, 1 and 0.999... are two different ways of representing the same number and *not* two different numbers.
This is because one of the rules the real numbers follow is *trichotomy*, which says that if x and y are real numbers then exactly one of the statements x-y>0, x=y and y-x>0 must be true. (Depending on the axiomatization chosen trichotomy can be either an axiom or a theorem. Either way it is true.) The requirement in the problem that the numbers be different rules out the second possibility.
In fact we can make an even stronger statement. There is a basic theorem (called the Archimedean principle) which makes an even stronger assertion, given any two distinct reals there is always a rational number between them. So let n/m be a rational number between 0 and x-y. Then x and y must differ by more than 1/m. So you see that between any two real numbers there is always a finite visible gap. There is therefore no such thing as an infinitesmal in the standard real number system.
(Google will provide adequate references to demonstrate that I'm not just making this up.) |
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