Finding the probability of a tail event

Question

Let $(X_n)_{n \geq 1}$ independent for which we have $X_n \sim \exp(\lambda_n)$ with $\lambda_n:=\log(n+1)$.

For $p>0$ consider the event $E_\rho=\left\{X_n\geq\rho \text{ for infinitely many } n\geq 1\right\}$.

I know that $P(E_{\rho})$ is either $0$ or $1$ by Kolmogorov's zero-one law (because $E_\rho$ is a tail event).

But how do I proof that $P(E_\rho)=0$ if $\rho>1$ and $P(E_\rho)=1$ if $\rho \leq1$?

Answer 1

For a fixed $\rho$, let $A_{n,\rho}:=\{X_n\geqslant \rho\}$. Then $E_\rho=\limsup_{n\to +\infty}A_{n,\rho}$. By the Borel-Cantelli lemma, $\Pr(E_\rho)=0$ if $\sum_{n\geqslant 1}\Pr(A_{n,\rho})$ converges and $\Pr(E_\rho)=1$ if $\sum_{n\geqslant 1}\Pr(A_{n,\rho})$ diverges. Now it remains to find upper and lower bound for $\Pr(A_{n,\rho})$.