Probability of $\limsup[X_n \geq n]$

Question

Let $X_1,X_2,\cdots$ random variables at $(\Omega,\mathcal F, P)$ such that $E(X_i)=0$ and $E(X^2_i)=1$ for all $i$. Prove that $P(\limsup[X_n \geq n]) = 0$.

Is my proof right?

Answer 1

We have that

$$\sum_{n=1}^{\infty} P[X_n \geq n] \leq \sum_{n=1}^{\infty} \frac{Var[X_n]}{n^2}= \sum_{n=1}^{\infty} \frac{1}{n^2}< \infty.$$

(I used the Chebyshev's inequality)

Hence, by Borel-Cantelli

$$P(\limsup[X_n \geq n]) = 0.$$

Q.E.D.