Let $f$ be a holomorphic function in point $z$. Prove that the inequality $|f^{(n)}(z)|>n^{2n}$ isn't true for every n.

Question

I am told that the conclusion as to why this is, is derived from the following theory, but I just don't see how, and was hoping someone could see. The theory goes like this:

If $f$ is holomorphic/analytic on some $\Omega$, which in it's inside contains a disk $D(a,r)$ with it's center being $a$, radius $r$, then Cauchy's inequality apply's:

$$|f^{(n)}(a)|\leq\frac{Mn!}{r^n}, (n=0,1,2...)$$ where $M$ is a const. such that $|f(z)|<M.$

Answer 1

Hint: prove that $n^{2n}r^n/n!$ is not bounded.