Prove $-1$ and $1$ are the only units in $\mathbb{Z}$

Question

Prove $\mathbb Z^*=\{-1,1\}.$

I have a proof, which is posted as an answer below. I'm looking for an alternate proof.

Answer 1

Suppose $ab=1$. In particular $a,b \neq 0$. Note that you have $|ab| \geq |a|$ and $|ab| \geq |b|$ since we are working with integers (you can prove easily by induction on $m$ that for all $n,m \in \mathbb{N}\setminus \{ 0 \} $ you have $ nm \geq n$).

So $1 = |ab| \geq |a|$. This means that $a \in \{ -1, 0, 1\}$.

This means that $\mathbb{Z}^* \subseteq \{ -1, 1\}$. Clearly, the opposite inclusion is trivial.

Answer 2

If $x$ is a unit $\mathbb{Z}$, then there is one $y \in \mathbb{Z}$ such that $xy = 1.$ So finding the units in $\mathbb{Z}$ is the same as finding the integer solutions of the equation $xy = 1.$