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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).

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    Thank you all for your $c$omment$s$. @alex.jordan, $s$ome research following your comment has lead me to believe that such sequences are called "asymptotically periodic". I will post an answer below.2011-12-18

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A sequence with the described property is called an asymptotically periodic sequence.

A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence $x_1,x_2,x_3,\dots$ is asymptotically periodic if there exists a periodic sequence $a_1,a_2,a_3,\dots$ for which: $\lim_{n\to\infty}x_n - a_n = 0$

Source: http://en.wikipedia.org/wiki/Periodic_sequence#Generalizations

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    +1 for the wikipedia reference. But, in analogy to this terminology, one might call a convergent sequence *asymptotically constant*, which I do not like very much. :-)2011-12-18