Let $(z_n )_n∈N$ be a sequence of non-zero complex numbers such that $|z_n |$ → ∞ as n → ∞. Let r > 0. Show that the sequence of functions $\prod_{k=0}^n E_k (\frac{z}{zk} )$ converges uniformly on $\overline B(0, r)$, the closed ball of radius r.
Any help is greatly appreciated!
Edit: Sorry, I neglected to mention that $E_k(z) = (1 − z)e^{z+\frac{z^2}{2}+...+\frac{z^k}{k}}$ .