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

if $A = \mathbb{N}$ then the double-indexed set would be like an "infinite square matrix".
Hope you can get a vague picture of what a double-indexed set is like.
Roughly speaking this proposition is like saying that the "biggest" one among the "smallest" ones in each "column" is smaller than the "smallest" one among the "biggest" ones in each "row". It is an "infinite version" of what is quite obvious in a finite matrix.
