Show that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|<1$.

Question

1) Show that $f_k(z)=z^k/k$ converges uniformly for $|z|<1$

2) Show that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|<1$.

My Try:

I did part 1. In part 2, I can prove that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|\leq 1$. But how can I prove it for $|z|<1$? Can anybody please help me to figure it out?

Answer 1

HINT:

$$\lim_{k\to \infty}\left(1-\frac1k\right)^{k-1}=e^{-1}\ne 0$$