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I need to show that $F(x)=\exp(-x-\sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_{k} >0, b_{k} \in \mathbb{R}, k \in \mathbb{N}$ with $\lim\limits_{k \to \infty} \frac{U(kx)-b(k)}{a(k)}=D(x)$ where $D(x)=G^{\leftarrow}(e^{\frac{1}{x}}),$ $G$ is a nondegenerate distribution function and $U=(\frac{1}{1-F})^{\leftarrow}$. As a hint i know that the following holds $$ \lim\limits_{k \to \infty} U(n_{k}x)-\log(n_{k})= U_{1}(x)$$ where $U_{1}$ is the inverse of $\exp(x+\sin(x))$ and $n_{k}=[\exp(2\pi k)].$ Any approach?

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