2012 Putnam B4

Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$

have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)

假设 $a_0 = 1$ 且 $a_{n+1} = a_n + e^{-a_n}$($n=0,1,2,\dots$)。 $a_n - \log n$ 是否

有一个有限的限制为$n \to \infty$？ (这里 $\log n = \log_en = \ln n$。)