Suppose there exists an r' such that for some real number r, p(r)=r', where p(r) gives the real number that precedes r. What does that mean? Does it mean there are no numbers between r and r'? Because that's what I think predecessor means, even though it's trivial to prove that there are numbers between r' and r. For example, (r'+r)/2, the average of r and r', is between the two numbers. And there are also numbers between r and (r'+r)/2, and between r' and (r'+r)/2. So there can't possibly be a real that is the predecessor to another real, because there are always more real numbers between any two reals.