Suppose $X$ is a random variable such that for any $\alpha>1$, we have $$\lim_{n \to \infty}\frac{\mathbb{P}(X \geq \alpha n)}{\mathbb{P}(X \geq n)}=0$$. Then prove that $X$ admits finite moments of all orders.
$$$$My attempt is as follows : We know that if $\mathbb{E}[|X|^s]<\infty \implies \mathbb{E}[|X|^r]<\infty$ for all $r<s$. So if we show that all the even order moments exist then we are done. But I am stuck at this point.