I'm just starting to learn how to use proofs by contradiction, and I am just wondering if this works.
Theorem (Archimedean Property): If $a$ and $b$ are any positive integers, then there exists a positive integer $n$ such that $na \geq b$.
Proof:
Suppose not. Suppose that there exists a positive integer $a$ and $b$ such that for all positive integer $n$, we have $na<b$. Let $n=b$, so $ab<b\Rightarrow \frac{ab}{b}<\frac{b}{b}\Rightarrow a<1.$ Thus, $a$ is a positive integer and $a$ is less than $1$, which is a contradiction.