Given that
1) $E=\{e_1,e_2,...,e_k\}$ is a finite set.
2) $\bar a>0$ is some finite constant so that $[0,\bar a]$ is a compact set.
Show that the following space of sequence of functions is compact in some topology.
$\mathcal{F}=\big\{\{f_t\}_{t=0}^{\infty}|f_t:E^{t+1}\rightarrow[0,\bar a]\big\}$
I have some background in analysis, but assume that I'm a complete noob about topology, which I kind of am, so I need clear steps of the proof. I was told I should probably use Arzela-Ascoli and/or Tychonoff's Theorem at some point.
Thanks in advance!