Weierstrass elementary factors

The Weierstrass elementary factors $E_n$ are defined for $n \in \{0,1,2,\ldots\}$ by $$E_n(z)=\left\{ \begin{array}{ll} 1-z &; n=0 \\ (1-z)e^{z+\frac{z^2}{2}+\frac{z^3}{3}+\ldots+\frac{z^n}{n}} &; \mathrm{otherwise}. \end{array} \right.$$

Properties

Proposition: The following formula holds for $|z| \leq 1$: $$\left| 1-E_n(z) \right| \leq \left| z \right|^{n+1}.$$

Proof: █