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floor(x) = m iff m ? x < m+1 and m is an integer. floor: R->Z is well defined along all its domain since the Archimedean Property guarantees us that for all x in R, there exists m in Z s.t. m ? x < m + 1. (definition of floor function)
Suppose floor(x) = m and let y > x. Then we have m ? x < y, so either y < m + 1 or m + 1 ? y. First case implies that f(x) = f(y); second case implies that there is n > m such that n ? y < n + 1 (by the Archimedean Property again) and therefore floor(y) = n > m = floor(x). By definition of monotonocity, the floor function is monotonous non-decrescent; we have just proved that x < y implies floor(x) ? floor(y).
