$$f:N \rightarrow B,\space B\subset N $$ and $B$ is finite, $S$ is the sequence constructed by $f(1),f(2)\cdots f(i)\cdots $.
 Now, if $f$ is a computable function,is $S$ eventually periodic?


Update: Secondly, if the computable function is computable in p time, is the $S$ eventually periodic? Or, under which computational complexity of the computable functions is $S$ eventually periodic?or such a computational complexity condition of computable function does not exist?
Any reference is welcome.


Thirdly, what computational complexity of computable functions is the boundary under or above which the $S$ is eventually periodic or is not periodic ultimately?



If the set of the output of a computable function is finite, is the sequence periodic eventually?
Regarding the 2nd question, the set of output sequences of an [autonomous finite automaton](https://books.google.com/books?id=s58EXPW6B6sC&pg=PA215&lpg=PA215&dq=output%20of%20finite%20automaton%20periodic&source=bl&ots=Liwgupg94l&sig=IKEczqfQqDhR6QZ7-CfG4YV57FE&hl=en&sa=X&ved=0ahUKEwjUz8Ody53QAhWIi1QKHeEwB8UQ6AEIIzAB) consists of ultimately periodic sequences.


