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
 A: You have a bunch of sequences, call them Sequence A, Sequence B, and so on; or as the question says, Sequence $\alpha$ for all $\alpha\in A$.
The left-hand side takes the liminf of each sequence, to give a function $L(\alpha)$.  Then you take the supremum of all the $L(\alpha)$.
The right-hand side forms a new sequence: Take the supremum of the first elements; the supremum of the second elements; and so on.  Then take the liminf of the new sequence.
A: Being indexed by two sets $X$ and $Y$ is effectively (or literally, depending on your definitions) the same as being indexed by the product space $X \times Y$. Your intuition, however is dead on. Just think of it that for each $a\in A$, you have a sequence.
Actually solving the problem more or less involves writing out the definition of supremum and playing around for a while. 
A: 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.
A: $n$ indexes the sequence {$x_n$} for given $\alpha$, where $\alpha$ is used to index all such sequences for some appropriate indexing set $A$.
