Apologies in advance if this is a stupid question with a trivial answer. I don't know much topology and yet, I need to prove this result rigorously.
Assume time is discrete, $t\in\{0,1,2,...\infty\}$. $E=\{e_1,...,e_n\}$ is a finite and discrete set and let $(E,\mathcal{E})$ be the corresponding measurable space. Also let $(E^t,\mathcal{E}^t)$ be the measurable space defined on the Cartesian product.
$a_t$ is a discrete-time stochastic process with the following properties:
1) $a_0$ is a given constant.
2) $a_{t+1}$ is measurable with respect to $\mathcal{E}^t$.
3) $a_t\in[0,\bar a]$ for all $t$, and $\bar a>0$ is a constant.
The result I need to prove is that the set of all such stochastic processes is compact (in some topology). I'm pretty sure that the statement is correct, but I need a step by step rigorous proof. I initially assumed this result holds by Tychonoff's theorem, but apparently this argument doesn't make sense and I need to invoke Arzela-Ascoli theorem, which I have no idea about. Any help would be greatly appreciated.
I'm going to eventually use this result in conjunction with Berge's Theorem of Maximum and the set above happens to be the constraint set in the optimization problem.