Chapter 13.3, Problem 14E

Show that if $M=\left(S,I,f,s_o,F\right)$ is a deterministic finite-state automation and $f\left(s,x\right)=s$ for the state $s\in S$ and the input string $x\in I^{*}$ , then $f\left(s,x^n\right)=s$ for every nonnegative integer n. (Here

   $x^n$ is the concatenation of n copies of the string x, defined recursively in Exercise 37 in Section 5.3.)