Proving something about an object by expressing it as an ultraproduct of easier to understand objects I was looking at this proof of the Ax-Grothendieck theorem, the theorem that any injective polynomial function on $\mathbb{C}^{n}$ is surjective. The conclusion of the Ax-Grothendieck theorem is, in the relevant sense, essentially first-order expressible in the structure $\mathbb{C}$ or any other field $F$; it holds if $F$ satisfies the first order sentence that any injective polynomial function on $F^{n}$ with coordinates of degree at most $m$ is surjective, for any $n$ and $m$. So, by Łoś's 
theorem we can prove this conclusion for $F=\mathbb{C}$ if we can prove it for $F$ an algebraic closure of a finite field, which we a priori understand better than $\mathbb{C}$, and show that $\mathbb{C}$ is the (nonprincipal) ultraproduct of these algebraic closures.
My question is whether there are other proofs like this. Where else can we prove that an object $A$ has a given property $P$ that, viewing $A$ as a structure for a language $\mathcal{L}$, is essentially first-order expressible in $\mathcal{L}$, by expressing $A$ as the ultraproduct of easier to understand objects already known to have property $P$?
 A: In general, it's hard to show that a structure $M$ of interest is actually isomorphic to an ultraproduct $\prod_{i\in I} M_i/U$. In the case of $\mathbb{C}$, we can use the facts that $\mathbb{C}$ and $\prod_{p}\overline{\mathbb{F}_p}$ are both algebraically closed fields of characteristic $0$ and cardinality continuum, and there is a unique such field up to isomorphism (i.e. the theory $\text{ACF}_0$ is $2^{\aleph_0}$-categorical). 
But we don't need isomorphism - it suffices to show that our structure $M$ is elementarily equivalent to the ultraproduct $\prod_{i\in I} M_i/U$. In the case of $\mathbb{C}$, we only need to show that $\mathbb{C}$ and $\prod_{p}\overline{\mathbb{F}_p}$ are both algebraically closed fields of characteristic $0$ and use the fact that $\text{ACF}_0$ is complete. This argument is obviously more broadly applicable (there are more complete theories than uncountably categorical theories!), so most applications of Łoś's theorem of the kind you're interested in will have this form.
Another classic application of Łoś's theorem is the Ax-Kochen theorem. Ax and Kochen phrased their argument differently, but in modern language, two Henselian valued fields of residue characteristic $0$ are elementarily equivalent if and only if their residue fields and value groups are elementarily equivalent. As a consequence, we have the elementary equivalence $$\prod_{p} \mathbb{Q}_p/U \equiv \prod_p \mathbb{F}_p((t))/U$$
for any non-principal ultrafilter on the set of primes. This theorem makes precise the intuition that the $p$-adic fields and the formal Laurant series fields over $\mathbb{F}_p$ have similar behavior "in the limit $p\to\infty$".
The algebraic consequence (often called the Ax-Kochen theorem) is the following: For each degree $d$, there exists a finite exceptional set of primes $P_d$, such that for any $p\notin P_d$, any homogeneous polynomial over $\mathbb{Q}_p$ of degree $d$ in at least $d^2+1$ variables has a non-trivial zero in $\mathbb{Q}_p$. 
Proof: A field is called $C_k$ if any homogeneous polynomial of degree $d$ in $d^k+1$ variables has a non-trivial zero (this easily implies that any homogeneous polynomial of degree $d$ in at least $d^k+1$ variables has a non-trivial zero). Algebraically closed fields are $C_0$, and $C_1$ fields are called quasi-algebraically closed - these include the finite fields $\mathbb{F}_p$. Lang proved that $\mathbb{F}_p((t))$ is $C_2$ for all $p$. Now the $C_2$ condition is expressible by a schema of first-order sentences, one for each degree $d$. And the conclusion follows by the elementary equivalence above and two applications of Łoś's theorem.
