note
tilly
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 <i>not</i> two different numbers.<p>
This is because one of the rules the real numbers follow is <i>trichotomy</i>, 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.<p>
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.<p>
(Google will provide adequate references to demonstrate that I'm not just making this up.)
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