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 $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.
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 $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.