"standard" proof that a compact field is finite

This is a proof that a compact field is finite, so you might want to read that first. To help you compare this proof to the NSA proof that a compact field is finite (which uses non-standard analysis), I have endeavored to use similar terms in both.

Let F be a compact field, and suppose to the contrary that F is infinite. Take an infinite sequence of different elements of F. Applying compactness, we see that that sequence must have a subsequence y_n which converges to a limit. Denote x = lim_n→∞y_n. Define z_n = y_n-x. Then (subtraction is continuous in a topological field) lim_n→∞z_n = 0.

Now define w_n=z_n^-1 (the sequence of multiplicative inverses of the z's). F is compact, so there exists a convergent subsequence of w_n: say v = lim_k→∞w_nk exists, for some increasing sequence of indices n_k. By continuity (recall that z_nk→0),

0*v = lim_k→∞ z_nk*w_nk = lim_k→∞ 1 = 1

So our assumption that F is infinite cannot hold; a compact field F must be finite.

The other proof is probably "simpler". Except that you need to know more logic, to work with non-standard analysis...