Baby rudin theorem 8.8 how he found the two inequalities (56 and the last)?

Thanks!

The last inequality is achieved as follows.
$ |Q(re^{i\theta})|$
$=|1+b_k r^k e^{ik\theta}+b_{k+1}r^{k+1}e^{i(k+1)\theta}+ \dots +b_{n}r^{n}e^{in\theta}|$
$\le|1+b_k r^k e^{ik\theta}|+|b_{k+1}r^{k+1}e^{i(k+1)\theta}|+ \dots +|b_{n}r^{n}e^{in\theta}|$
$=1-r^k |b_k|+|b_{k+1}|r^{k+1}+ \dots +|b_{n}|r^{n}$
$=1-r^k$ {$|b_k|-r|b_{k+1}|-\dots-r^{n-k}|b_{n}|$ }
