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
 A: 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.
A: The infimum of $X_{n}$ being smaller than a is equivalent then at least one of the $X_{n}$ being smaller than a. So an element is in the first set, iff it is in at least one of the sets the union is operating on. So the set is the union.
The argument for sup is basically the same. 
The second question depends on your definition of the limsup, because the equality holds, it is possible to just define the limsup like this. In order to derive the equality, i need your definition.
