Is this proof of the convergence of a recurrence relation correct?

Question

The relation: \begin{equation} a_{n+1} = a_n + e^{-a_n} (a_0 = 0) \end{equation} Does it converge or diverge?

I've been working on this on and off for about a year, and I think I've figured out a proof:

Proof:

You can easily rewrite the relation as the following sum: \begin{equation} a_{n+1} = \sum_{m=0}^{n}e^{-a_m} \end{equation} Thus the question of convergence of divergence boils down to figuring out if the following series converges or diverges: \begin{equation} a_{n+1} = \sum_{m=0}^{\infty}e^{-a_m} \end{equation} Now Raabe's Test states for an arbitrary series: \begin{equation} \sum_{m=0}^{\infty}b_m \end{equation} If: \begin{equation} \lim_{n\to\infty}\left|\frac{b_n}{b_{n+1}}\right| = 1 \end{equation} And setting: \begin{equation} R = \lim_{n\to\infty}n\left(\left|\frac{b_n}{b_{n+1}}\right|-1\right) \end{equation} If $R > 1$, that series converges. If $R < 1$ that series diverges.

Now applying that test to my relation: \begin{align*} \lim_{n\to\infty}\left|\frac{e^{-a_n}}{e^{-a_{n+1}}}\right| &= \lim_{n\to\infty}\frac{e^{a_{n+1}}}{e^{a_{n}}}\\ &= \lim_{n\to\infty}\frac{e^{a_{n} + e^{-a_n}}}{e^{a_{n}}}\\ &= \lim_{n\to\infty}e^{e^{-a_n}}\\ \end{align*} Assume $a_n$ diverges, i.e: \begin{equation} \lim_{n\to\infty}a_n = \infty \end{equation} Since: \begin{equation} \lim_{n\to\infty}e^{-n}= 0 \end{equation} Thus: \begin{equation} \lim_{n\to\infty}e^{e^{-a_n}}= e^0 = 1 \end{equation} Thus it fulfills the first condition of Raabe's Test. Now set $R$: \begin{equation} R = \lim_{n\to\infty}n\left(\left|\frac{a_n}{a_{n+1}}\right|-1\right) = \lim_{n\to\infty}n\left(e^{e^{-a_n}}-1\right) \end{equation} By a property of limits: \begin{equation} \lim_{n\to\infty}n\left(e^{e^{-a_n}}-1\right) = \lim_{n\to\infty}ne^{e^{-a_n}} - 1 \end{equation} And by the fact that the limit of a product is the product of the limits: \begin{equation} \lim_{n\to\infty}n\lim_{n\to\infty}e^{e^{-a_n}} - 1 = \lim_{n\to\infty}n - 1 \end{equation} Which is a divergent limit i.e: \begin{equation} \lim_{n\to\infty}n - 1 = \infty \end{equation} Therefore: \begin{equation} R > 1 \end{equation} Therefore the series converges. But since we assumed the series diverges, we have reached a contradiction and thus the series converges.

Is this proof correct?

Now please, I beg of you, could you please just point out the mistakes in my proof, and not solve the problem? This is a problem I work on when I'm bored or when I have free time to so pass time. So if my proof is incorrect, and you solve the problem, I'll have no problem to while my time away.