How do you show that there is an integer greater than any given real number? The real question is, given the axioms for the real numbers (Dedekind-complete ordered field), how do you define the "real integers" within them. Two possibilities spring to mind. Definition A: The "real integers" are the minimal subset of R that contains 1 and is closed over negation and addition. Definition B: 1 is a "real integer", and there is a unique "real integer" in every half-open unit interval of R. Definition A seems more in the spirit of the number theory definition of Z. However, I do not immediately see how it answers my original question. Definition B immediately answers my original question. For any real number r, there is a unique integer on the half-open interval (r, r+1] which must be greater than r. Presumably definitions A and B are equivalent. Is the proof easy?