Real number comparisons: must a number be less than or equal to or greater than another number? I've been reading Knuth's Surreal Numbers recently and came up with this question about real numbers.
Is is true that among all three relationships (=, >, <), a real number must be of one, and only one relationship with another real number. If this is true, how to prove it?

Yes - this is called the trichotomy property (
 http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 ) . It can be easier to see depending on how you define the real numbers. For instance: http://en.wikipedia.org/wiki/Dedekind_cut (see ordering of cuts part way down the page)
http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29
