An inequality, comprising of liminf and Sup of a set The following was an example in a book called Applied analysis by Hunter. I am not exactly sure of what it really means or to how to approach it
{ $ x_{n,\alpha}\in \mathbb R $ |$ n \in \mathbb N $ and $\alpha \in A $} is a set of real numbers indexed by the natural numbers and a set A
$ \sup_{\alpha \in A}$[$\liminf_{n \to \infty } x_{n,\alpha}$] $\le$  $\liminf _{n \to \infty} [\sup_{\alpha \in A} x_{n,\alpha}$]
I have some problems understanding the set too, what does it mean to be indexed by both $\Bbb N $ and A? Is it just a different sequences indexed by the elements of A? If so how can I approach the problem.
Thank you

$n$ indexes the sequence {$x_n$} for given $\alpha$, where $\alpha$ is used to index all such sequences for some appropriate indexing set $A$.
