On the asymptotic behaviour of one-sided large deviation probabilities

Let $X_1,X_2,\dots$ be i. i. d. random variables,
$$ \mathbf EX_1=0,\qquad F(x)=\mathbf P\{X_1<x\},\qquad S_n=X_1+\dots+X_n $$
We prove that if for $x\to\infty$
$$ 1-F(x)\thicksim x^{-\alpha}h(x),\qquad \alpha>1, $$
where $h(x)$ is a slowly varying function, then
$$ \mathbf P\{S_n\ge x\}\thicksim n(1-F(x))\qquad\text{for}\ n\to\infty\ \text{and}\ \liminf_{n\to\infty}x/n>0. $$