show that finite orbit has a periodic point

Question

i am trying to prove that a finit orbit has a periodic point. The true statement of the exercise is: "prove or disprove that if a point x has a finit orbit, then x is either a periodic point or a eventually periodic point."

Thank you very much

Answer 1

Thank you very much!!,

i use cycle detection. so i build this:

For a function f that map a finite set S to itself and a point x0 in S and considering the sequence x0, x1=f(x0), x2=f(x1),...,xn=f(x_n-1) for n>1, eventually must be equal for any previous term in the sequence. Let k be the smallest index. Let j<k such that xj=xk then xj has an orbit of period k-j.

I am new here, so sorry i a make some mistakes in formats, etc... Thank you very much!