Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8): 

Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a field $k$. Then $H^{i}(X,F)$ is finite for $i\geq0$.

He deduces this via Hochschild-Serre from the statement, that on the big étale site of $X$, constructible sheaves are stable under higher direct images of proper Morphisms (p.223, Theorem VI.2.1).


My Question is: Is there a "basic" proof of the proposition, which doesn't involve other Grothendieck topologies than the small étale sites (and possible the Zariski-topology)?

I would like to use this for my Graduation thesis, so if you think this is easy and should be a standard exercise and I should solve it myself - tell me, I will try again.

Thanks!