Does this recurrence relation of exponentials diverge?

Question

Does the following recurrence relation diverge? $$x_{k+1} = x_k - \frac{e^{-x_k}}{(1+e^{-x_k})^2}$$ where $x_0 <0$.

I have plotted the first one million steps of the sequence using $x_0=-1$ alongside $-\log(k)$ and they look almost identical, suggesting that $x_k$ does indeed diverge.

$-log(k)$" />

Answer 1

Assume $x _n \to x $. Then $$0 = -\frac{e^{-x}}{(1 + e^{-x})^2} \implies e^{-x} = 0 $$ so we get a contradiction. Hence there $\not\exists x \in \mathbb{R} $ such that $x_n \to x$. On the other hand , $x_n$ is a decreasing sequence, so the only possible outcome is that $x_n \to -\infty$.