Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$. I am trying to prove $|e^z - e^\omega| \leq |z - \omega|$ for $z,\omega\in\{z\in\mathbb{C}\;:\;\Re(z)<0\}$ and I get stuck in an inequality of the stated in the title : $|e^x + e^y| \leq |x - y|$ for $x,y<0$.
This is what I have tried:
\begin{equation}\left|e^z - e^\omega\right| = \left|e^{i\Im(z)}\right|\left|e^{\Re(z)} - e^{\Re(\omega)}e^{i(\Im(\omega) - \Im(z))}\right| \leq \left|e^{\Re(z)} + e^{\Re(\omega)}\right|
\end{equation}
Is it true that $|e^x + e^y| \leq |x - y|$ for $x,y<0$? If yes, any help will be appreciated.
Thank you!

The inequality $|e^x + e^y| \leq |x - y|$ is not true for all $x,y <0$.
Suppose that we have  $|e^x + e^y| \leq |x - y|$  for all $x,y <0$.
With $x \to 0$ and $y \to 0$ we would get $2 \le0$. !
