We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote $$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$ By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.
We suppose that the property $\mathcal{P}$:
$$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$
is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote
$$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$
By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.