Simple Proof Methods

It's time to show that there exist infinitely many integers by using a proof by contradiction!

Theorem. [Infinity of $\mathbb{Z}$] There exist infinitely-many integers. That is, $|\mathbb{Z}| = \infty$.

Proof. To obtain a contradiction, assume that there exists a finite number of integers. Since all those integers have different values, one of these integers must be the largest one; let's call it $n$.
But hey: note that $n + 1$ is an integer. Moreover, note that $n + 1 > n$, which means that $n + 1$ is larger than $n$.
This is a contradiction because we said that $n$ was the largest of those finitely many integers, which means that $n + 1$ is a brand new integer outside of the finite integers that we assumed. In other words, no matter which finite list of integers you pick, there's always going to be a bigger integer.

This shows that our assumption that there exists a finite number of integers is wrong, so it must be true that there exist infinitely-many integers.◼