Maybe this is extremely simple, but i havent found a specific answer for this online.
For a sequence of independent continuous random variables $X_n$ ,$n=1,2,3,...$ , all with the same probability density function $f(x)>0$ for all real $x$, and a specific real number $b$, what do the following events mean in simple terms and what is the probability of each?
a)$\lim_n\sup\{X_n \gt b\}$ b)$\lim_n\inf\{X_n \gt b\}$ c)$\lim_n\sup\{X_n \le b\}$ d)$\lim_n\inf\{X_n \le b\}$
Is $lim_nsup E_n$ the same as what is usually written $\underset{n\to\infty}{lim sup}E_n$? If so, for events $E_n$ the definition is $lim_n sup_{k\ge n} E_k$, where $sup$ is interpreted as union. For the limes inferior, the definition is $lim_n inf_{k\ge n} E_k$, where inf is interpreted as intersection. These are then, respectively decreasing and increasing chains of sets (unions getting smaller, and intersections larger, as you range over fewer sets). To answer your question, the usual explanation "in simple terms" is that the lim sup gives those $\omega$ that are in infinitely many of the $E_n$, and the lim inf gives those $\omega$ that are in all but finitely many.
The probabilities you mention will depend on the distribution of $X_n$. E.g. if $P[X_1>b]=p>0$, one borel-cantelli theorem implies $P(lim sup_n \{X_n>b\})=1$. If $p=0$, the other borel-cantelli implies the lim sup will be 0.