Problem Elsewhere, I have seen it proposed by a mathematician that there is a "greatest number", as opposed to infinitely many numbers. This number is supposed to be exceedingly large, larger than Graham's number or other such "big" numbers, yet still finite. I dislike this notion immensely, and so have a rough proof that no such greatest number exists. This is outlined below.
Proof Assume that there is some such greatest number $z$. If such a number $z$ exists, then $\frac{1}{z}$ must also exist, and be the smallest, non-zero positive number. Because this number can be expressed as a fraction, $\frac{1}{z}\in\mathbb{Q}$, we now encounter an issue. If there is a greatest number, then there cannot be any irrational numbers, as there is a smallest non-zero number. There are various proofs that there are irrational numbers, and so that is not included here. Because of this contradiction, irrational numbers must exist, yet cannot exist, there is a contradiction, and so the assumption that there is a biggest number $z$ must be false. Therefore there is no biggest number.
Request Is this proof correct? Is there some logical error or unknown construct which invalidates the proof? If not, is there a better proof?
The closest proof I could find is that $\mathbb{N}$ has infinite elements on proof-wiki, but this is not quite the same issue.