I am having difficulties proving the following: If $\cal{A}$ is a connected class (i.e $\cal{A_\emptyset} = \emptyset$), then I need to show that the powers of $\mathcal{A}$ form a locally finite sequence. In other words, the sequence $\mathcal{S} = (\mathcal{A}^0, \mathcal{A}^1, \mathcal{A}^2, ...)$ is locally finite.
I know that to show that $\mathcal{S}$ is locally finite, i need to show that for any fintie set $X$, there are at most a finite amount sets $[(\mathcal{A}^i)_{X} : i \ge 0]$ that are not empty. However, I am quite unsure as to how to start.
If I could have a hint on how to do this, it would be greatly appreciated. I should also add that I don't have any other theorems to work with, besides for the basic definitions provided.
Thanks!
EDIT: I would also like a reference to material like this, if possible. I have no idea what this would be classified as though. We are seeing this in my enumeration course, but searching on Google does not seem to yield much information. On Wikipedia, I found a list of structures/classes (http://en.wikipedia.org/wiki/Algebraic_structure), though I am unsure as to what I should be looking up.