It appears that the following holds: $$1-e^{-e^{-c}} \sim e^{-c}$$
I do not see why this should hold. How do I prove it? I would be very happy to see multiple ways of proving this, as it is always good to have more tools in one's mathematical toolbox.
Moreover, it seems to me that the ratio between these to value tends to 1 rather quickly (probably exponentially). Is there a way how to say something about the speed of convergence?
I'm assuming that this holds as $c\rightarrow +\infty$. In that case, $e^{-c}\rightarrow 0$ and $$\begin{split} 1-e^{-e^{-c}} &= 1-(1-e^{-c}+\mathcal O(e^{-2c}))\\ &= e^{-c}+\mathcal O(e^{-2c})) \end{split}$$