We propose to show by contradiction the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$ We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that: $$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$
We propose to show by contradiction the property $\mathcal{P}$:
$$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$
We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that:
$$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$