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$\large \begin{cases} a_0 = 1 \\ a_{n+1} = a_n + e^{-a_n} & \text{for } n = 0, 1, 2 ... \end{cases}$

Let $a_n$ be defined as above, find $\displaystyle \lim_{n \to \infty} a_n - \log (n)$.

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