Deriving time-scaling property for Fourier Series thanks for taking the time to help with this problem!

I have to prove the time-scaling property:

$$
x_{(m)}[n] =
\begin{cases}
x[n/m], &amp; n=0,\pm m, \pm 2m,...\\
0, &amp; otherwise
\end{cases}
$$

First, for part (a), Show  that   $$x_{(m)}[n]$$ has  a  period  of $$ mN$$

Then, for part (b), Show  that   if $$\\x[n]=v[n]+w[n]\\$$ then$$\\x_{(m)}[n]=v_{(m)}[n]+w_{(m)}[n]
$$

There's a bit more to the problem after this, but I'm mostly concerned with these first two pieces. I know that for a periodic function $$x[n]=x[n+N]$$

Am I supposed to somehow prove that $$x[n/m+mN]=x[n/m]$$
using the Discrete Fourier Series analysis equation? What is a rigorous way to prove this. 

For part (b), I'm unsure of how to prove the linearity in a way that doesn't seem trivial.

Thanks!
It looks more like zero padding to create up-sampling resolution to me. 

Prove
x[n/m+N]=x[n/m], since your argument is now n/m and not n.