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

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.
