Originally Posted by
snowtea You already have everything.
The basic idea is you are using proof by contradiction.
You know for a fact that
$\displaystyle |S(a_0)| \leq |S(a_1)| \leq |S(a_2)|\leq\dots\leq |S(a_{2^n+1})|\leq2^n$
Now, assume that for no $\displaystyle m$ does $\displaystyle a_m\equiv a_{m+1}$.
This means the inequalities must be strict:
$\displaystyle |S(a_0)| < |S(a_1)| < |S(a_2)| < \dots < |S(a_{2^n+1})| \leq 2^n$
But this lead to a contradiction because this forces $\displaystyle |S(a_0)| < 0 $.
This means the initial assumption was incorrect, and there must exist an $\displaystyle m$ s.t. $\displaystyle a_m\equiv a_{m+1}$.