In 2.7 of this notes, it is shown that, using Borel-Cantelli lemma, if $E_n = \left\{\frac{X_n}{\log n} \geq 1\right\}$ then $\mathbb{P}(\limsup \ E_n) = 1$ but why does author conclude that if $Y = \limsup \frac{X_n}{\log n}$, then $\mathbb{P}(Y \geq 1) = 1$ ?
My question is Borel-cantelli lemma says events $\limsup$ of some events is 0 or 1, but in most books and online notes, an event in terms of random variable $\limsup X_n$ concluded to have probability 0 or 1. Can someone clarify the confusion ?
If $(a_n)$ is a sequence of real numbers such that $a_n \geq 1$ for infinitely many values of $n$ then $\lim \sup_{n \to \infty} a_n \geq 1$. Hence the event $(\frac {X_n} {\log n} \geq 1 \,\, \text {i.o.} )$ is contained in the event $(\lim \sup_{n \to \infty} \frac {X_n} {\log n} \geq 1)$.