I'm struggling to get my head round the mathematical proof for the alias frequencies in a sampled sine wave.
I understand that sampling a sine wave of frequency $f_0$ every $t_s$ seconds gives you:
$$x[n]=\sin(2\pi f_0nt_s)$$
I also understand that, because the sine wave is periodic every $2\pi$, you can add any multiple of $2\pi$ to the angle and get the same values for the sine, i.e.,
$$\sin(2\pi f_0nt_s)=\sin(2\pi f_0nt_s+2\pi m) \quad\text{(where $m$ is any integer).}$$
The proof I'm looking at then factors out $2\pi$ and $nt_s$ to get:
$$\sin\left(2\pi(f_0+\frac{m}{nt_s})nt_s\right)$$
...but then it says to let $m$ be an integer multiple of $n$ so we can replace the $\frac{m}{n}$ ratio with an integer $k$.
I don't understand how $m$ can go from being "any integer" to "an integer multiple of $n$". If $m$ is any integer and $n$ is an integer then how can the ratio between them be an integer?
I know I'm missing something obvious here and I'm searching for that light-bulb moment but it's not happening. Because this is so fundamental to DSP I don't just want to accept the formula and move on without thoroughly understanding it.