[Dr_Segfault]

What is your definition of "previous real number"? If you define p(r) such that p(r) < r and there does not exist any real number x such that p(r) < x < r, then it is easy to show that p(r) cannot exist using a proof by contradiction.

Given real number r, assume there exists a real number q such that q < r and there does not exist a real number x such that q < x < r. Let real number y = (q + r) / 2. It is trivial to show that q < y < r. Therefore we have a contradiction, and therefore there does not exist a real number q that meets our conditions for a "previous real number".

If you take issue with this, then I suggest you read up on the standard construction of the number systems from the naturals up to the reals. This is all very rigorously defined in terms of ZFC set theory.