On page 136 of Ergodic Theorem's by Ulrich Krengel, in the proof he sets: $c_n= \sup_j (n)^{-1} \sum_{i=0}^{n-1}x_{i+j}$ then he argues that for any $k,m$ positive integers one has: $$c_{km}\leq c_m$$
I don't see how this is true, obviously the factor 1/(km) makes it small, but then again there are more terms to be summed in $c_{km}$, i.e, $km-1$ while in $c_m$ only $m-1$ terms to be summed.
I don't see how this claim is true. Anyone?
I give you an idea for $k=2$, and I leave to you the general case.
We have: $$mc_m={\rm Sup}_j\{x_j+x_{j+1}+...+x_{j+m-1}\}$$ and $$2mc_{2m}={\rm Sup}_l\{x_l+x_{l+1}+...+x_{l+2m-1}\}$$
Fix an $l$, put $h=m+l$. We have:
$$x_l+x_{l+1}+...+x_{l+2m-1}=(x_l+x_{l+1}+...+x_{l+m-1})+(x_h+x_{h+1}+...+x_{h+m-1})\leq mc_m+mc_m=2mc_m$$ Hence $$2mc_{2m}\leq 2mc_m$$ and we are done.