Recently, I came across a statement which says the following: Suppose you have a binary sequence $(a_1,a_2,a_3,..)$ s.t there exists a fixed function $f:\mathbb Z_2^m\rightarrow \mathbb Z_2$ s.t $f(a_t,a_{t+1},..,a_{t+m-1})=a_{t+m}$ then the binary sequence is eventually periodic.
How do I prove that? dont think that it should be true but I can't come up with a counter example. Any suggestions?
There are only $2^m$ sequences of $m$ bits so by position $2^m+m$ you will see a repeat. Once you have a matching sequence the string will repeat because the next bit after the matching string will repeat and so on.