# $\limsup\frac{\log(X_n)}{c_n}=\frac{1}{a}$ almost surely

Let $$X_1,X_2,...$$ a sequence of random variables identically distributed following a Pareto distribution of parameter $$\alpha>0$$ such that $$P(X_1>x)=x^{-\alpha}$$ for $$x\geqslant 1$$. Show that for a sequence of constants $$c_1,c_2,...$$ we have:

$$\limsup\frac{\log(X_n)}{c_n}=\frac{1}{\alpha}$$ almost surely.

I thought of the following possible resolution however I am stuck:

I need to prove $$P(\limsup\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha})=0$$. By Borel Cantelli if I prove $$\sum_{n=1}^{\infty}P(\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha}))<\infty$$ then it implies $$P(\limsup\frac{\log(X_n)}{c_n}\neq \frac{1}{\alpha})=0$$

However I do not know how to make the computations.

Questions:

Is there something wrong with my reasoning? How should I solve this question?

Thanks in advance!