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

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. 
