State and prove the first and second Borel-Cantelli lemmas.

Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent Cauchy random variables. Thus, each $X_{n}$ is real-valued, with density function

$$f(x)=\frac{1}{\pi\left(1+x^{2}\right)} .$$

Show that

$$\limsup _{n \rightarrow \infty} \frac{\log X_{n}}{\log n}=c, \quad \text { almost surely, }$$

for some constant $c$, to be determined.