Why are these sets equal? how can I formally see that these following sets are equal? For $X_1,\ldots,X_n,\ldots$ random variables with values in $[-\infty,\infty]$, then:
$\{\inf_n X_n < a\} = \bigcup_n\{X_n < a\}$ and $\{\sup_n X_n>a\} = \bigcup_n
\{X_i > a\}$
I have also difficulties to see the following equality $\limsup_n X_n = \inf_m(\sup_{n\geq m}X_n)$.
Thank you for your help

Suppose $\inf_n X_n < a.$ Remember that "inf" means the largest lower bound.  That means nothing larger than that can be a lower bound.  Thus $a$ is not a lower bound of $\{X_n: n\}.$ To say that $a$ is not a lower bound of that set means $\exists n\  X_n<a,$ and that's the same as saying the event $\bigcup_n \{X_n<a\}$ occurs.
Conversely, suppose the event $\bigcup_n \{X_n<a\}$ occurs.  That means $\exists n\  X_n<a$.  That means $a$ is not a lower bound of $\{X_n : n\}$. And that implies all lower bounds are $<a$, since if some lower bound were $\ge a$ then $a$ would be a lower bound.  Hence the largest lower bound is $<a$, i.e. $\inf_n X_n < a.$
The argument for $\sup$ is the same with the inequalities inverted.
