Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$.
Show the existence of an integer $N > 0$ such that
$$\forall n \geqslant N , \quad f \left( \mathrm { e } ^ { - 1 / n } \right) \leqslant \frac { 2 } { 1 - \mathrm { e } ^ { - 1 / n } }$$