I recently had a discrete math midterm that I am trying to review, and one of the topics that really has me confused is the idea of proving uniqueness.
Let $x \in \mathbb R$. Prove that the $\lceil{x}\rceil$ is unique; that is, prove that there is at most one $n \in \mathbb Z$ with $n − 1 < x ≤ n.$
My professor, when doing these problems in the past, tended to use $x'$ as a second variable when proving uniqueness, but I am struggling with comprehending the logic behind these proofs. I understand the structure, i.e. that you are trying to prove that $x, x'$ both make the predicate true and that implies that $x=x'$, but I can't execute any of these proofs.
Also, please note that I am not trying to prove the existence here, but instead uniqueness.
Any help would be greatly appreciated.