Understanding the mathematical proof for the alias frequencies in a sampled sine wave
The reason is that if it is true for any $m$, it is also true for $m=kn$.

I will sketch the proof in another way.

Call $f_s = 1/t_s$ sampling frequency where $t_s$ is sampling period, the two signal $x(t) = \sin(2\pi f_0 t)$ and $x_k(t) = \sin(2\pi (f_0 + k f_s) t)$ have the same values at sampling instants (aliasing), i.e. $x[n] = x_k[n]$.

Indeed,

\begin{align}
x[n] &amp;= \sin (2 \pi f_0 n t_s) \\
x_k[n] &amp;= \sin \left(2 \pi (f_0 + k f_s) n t_s\right) \\
&amp;= \sin (2 \pi f_0 n t_s + 2\pi k f_s n t_s) \\
&amp;= \sin (2 \pi f_0 n t_s + 2\pi k n) = \sin (2 \pi f_0 n t_s)\\
&amp;=x[n]
\end{align}