I was given a proof of this in class, but it does not make sense to me, can someone provide a better proof, or explain the one I have provided, thanks.

Suppose $\gcd(a,n) = 1$, where $a$ and $n$ are natural numbers. Prove that $0$, $a$, $2a$, $3a,\ldots, (n-1)a$, is complete residue system modulo $n$.

Proof given in class:

\begin{align\*}
ka &\equiv ra \pmod{n}\\
&\Rightarrow k \equiv r \pmod{n}\\
&\Rightarrow \text{$k-r$ divides $n$}\\
&\Rightarrow \text{$k=r$ because $0\leq k\lt n$, $0\leq r\lt n$}
\end{align\*}