I am having trouble prooving something by induction. I understand the proof for $a_p$ but struggle ith $a_{p+1}$. Here is the extract of the book which should give a bit more insight on the problem 
I attempted it my self but I seem to have an extra term. Here is my work $$\begin{align} a_{p+1}&\le \sum_{s=1}^{p-1} \sum_{k+j=1}a_{k+s}b_{s,j}\\ &\le \sum_{s=1}^{p-1} \sum_{k+j=1}(p-1)K^2*KB^j\\&=K^3(p-1)\sum_{s=1}^{p-1} (B^0+B^1)\\&=K^3(p-1)^2+K^3(p-1)^2B\end{align}$$
Note that I have used the fact that $K<(p-1)K^2$ since p is an integer $>1$ hence $a_0,...,a_{p-1},a_p \le (p-1)K^2$
But according to the author we should have gotten $K^3(p-1)^2B$. Does anybody see where I made my mistake?