For any sequence s consisting of 1's and 2's, let r(s) denote the length of the nth run of same symbols in s .

There is a unique nontrivial sequence s such that $\displaystyle s(1) = 1$ and $\displaystyle r(r(s(n))) = s(n)$ for all $\displaystyle n$ :

$\displaystyle s = (1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, . . .)$

$\displaystyle r(s) = (2, 1, 2, 2, 1, 2, 1, 1, 2 , 2 ,1, 2, 2, 1, 1, 2, 1, . . .)$

Question :Prove or disprove that every segment of r(s) is a segment of s.

For example , the initial segment 1121 of s occurs in r(s) beginning at the 14th term.

P.S.

I am interested in hints (not full solution)