I'm studying proof in ergodic theory about the convergence in $L^2(\mu)$ of the averages of the form $$\frac{1}{N}\sum_{n=1}^NT^{p(n)}f$$ where $p(n)=n^2$ and $T^{p(n)}f=foT^{p(n)}$
Now in the proof i saw in the lecture notes this limit being used: $$\lim_{N \to +\infty}||\frac{1}{N}\sum_{n=1}^NT^{n^2}f-\frac{1}{r}\sum_{j=0}^{r-1}\frac{1}{[\frac{N}{r}]}\sum_{n=1}^{[\frac{N}{r}]}T^{(rn+j)^2}f||_2=0$$
Now in general if we have a bounded sequence $a_n$ then can someone explain to me why $\lim_{N \to +\infty}|\frac{1}{N}\sum_{n=1}^N a_{n^2}-\frac{1}{r}\sum_{j=0}^{r-1}\frac{1}{[\frac{N}{r}]}\sum_{n=1}^{[\frac{N}{r}]}a_{(rn+j)^2}|=0$.
I understand that if we take $r<N$ and apply the euclidean division theorem then we know that every natural number in $\{1,2....N\}$ has the form: $$rn+j,\text{ where }0 \leq j<r$$.
I tried to see my self, without success,why the limit is zero by taking $r=2$.
Can someone explain me,why this limit is zero?
Thank you in advance.