As the title states, I'm trying to prove $a_{n+1}=1-e^{-a_n}$ has no infimum if $a_1<0$. I'm not sure this is true, but calculating $a_n$ for large $n$ leads me to believe it is. I've already proved $a_{n}$ is monotone decreasing, and that if $a_{1}>=0$ then its infimum is 0.
I know the very basics of series and sequences; no Taylor and the such, but most limit calculation theorems as well as a decent amount (L'hopital, integrals, etc...) for calculating limits in $R$.
Any hints would be appreciated!