Borel-Cantelli lemma using to show lim sup

Question

Hello, I've got this exam I'm question. I'm practicing. I don't know how to start. obviously I have to get P(something i.o) = 0 or 1 and the use Borel-Cantelli lemma by proving the convergence or divergence of the sum.

Answer 1

Notice that we are taking a lim sup of random variables, not events. So Borel Cantelli cannot be directly applied.

We have $X_n \leq M$ a.s., so $\limsup _{n\to \infty}X_n \leq M$ a.s. (why?). So it suffices to show $\limsup_{n \to \infty} X_n \geq M$ a.s.

Consider the set $A_{\epsilon} = \{\omega : \limsup_{n \to \infty} X_n(\omega) \leq M-\epsilon\}$. It suffices to show $P(A_\epsilon)=0$ for all $\epsilon>0$. If $\omega \in A_\epsilon$, then there exists N such that for all $n\geq N$, $X_n(\omega)\leq M-\epsilon/2$. So, $$P(A_\epsilon) \leq \sum_{N=1}^\infty P( \forall n\geq N, X_n \leq M- \epsilon/2).$$ Now show that the right-hand side is zero using the definition of $M$.