[jamelk]

Your definition of f is circular: to calculate f(r) we need to know p(r), which in turn depends on f(r).

Cantor's diagonal argument shows that any mapping from the natural numbers to the real numbers must necessarily miss some real numbers out. It takes some time to get your head around if you aren't used to mathematical proofs, but it's definitely worth looking it up and trying to work through it if you're interested in this subject.

[bjourne]

p(r) is given a real number and retrieves its predecessor. Why is it circular?

[abcdefgxdf]

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.