a function $f$ is analytic in the disk $\Delta= D(0,1)$ and $|f'(z)|\leq c(1-|z|)^{-n-1}$ throughout $\Delta,$ where $n\in \mathbb N$ and $c>0.$ Show that $f(z)\leq cn^{-1}(1-|z|)^{-n}$ for $z \in \Delta$ under the condition $n \geq 1.$
I really don't know why $f'$ is bounded implies that $f$ is bounded. Maybe I can use Cauchy's estimate to do it, but I can not identify the way to attack this.
Thank you so much!