Consider sequences $(x_n)_{n=1}^\infty\subset\mathbb R$. Is there a name for the following property?
There exists $L\in\mathbb N$ such that:
$\lim\limits_{k\rightarrow\infty}x_{(kL+m)}=x^\ast_m$
for $m\in\{0,1,2,\dots,(L-1)\}$.
Here the $x^\ast_m$'s are not necessarily equal.
As an example, the sequence $x_n=(-1)^n +\frac n{n+1}$ has $x_{2n}\rightarrow 2$ and $x_{2n+1}\rightarrow 0$ as $n\rightarrow\infty$. In this case $L=2$ (choosing $L$ to be minimal).