The finiteness lemma states the following:
Fix an $o$-minimal structure $M$. Let $A\subseteq M^2$ be a definable subset, such that for every $x\in M$ $A_x=\{y|(x,y)\in A\}$ is finite. Then there exists $k\in\mathbb{N}$ such that $|A_x|<k$ for every $x\in M$.
My attempt:
I'm trying to approach the problem via the compactness theorem. Let $\phi$ be the formula which defines $A$, and take the contrapositive. That is, suppose that there is not a $k\in\mathbb{N}$ such that $|A_x|<k$ for every $x\in M$.
We note then that the set of sentences $\sigma_n: \exists x|A_x|>n$, for $n\in\mathbb{N}$ is true of $A$. One may show these are in fact sentences by rewriting the $\sigma_n$ as $$\sigma_n:\exists x\exists y_1...\exists y_{n+1}\bigg(\bigg(\bigwedge_\limits{1\leq i\leq{n+1}}\phi(x,y_i)\bigg)\wedge\bigg(\bigwedge_\limits{1\leq i<j\leq{n+1}}\neg y_i=y_j\bigg)\bigg).$$
Now, taking $A$, our definable set, we construct a structure $\mathcal{A}=(A,...)$ in the standard language of $o$-minimality (noting that $\mathcal{A}$ is not itself $o$-minimal). It follows easily that $\mathcal{A}\vDash\forall\vec{x}\phi$, and $\mathcal{A}\vDash\sigma_n$, by assumption of the contrapositive. We then note that the set of sentences $\forall\vec{x}\phi\cup\{\sigma_n\}_{n\in\mathbb{N}}$ is finitely satisfiable by $\mathcal{A}$, such that by compactness the entire set must be satisfiable by some model.
It is also clear to show that $A$ is itself a model of the infinite set of sentences (again by the contrapositive assumption), but this is akin to saying that there is some fiber $A_x$ which is infinite, thereby completing the proof by contrapositive.
--
I do not think this proof is valid. I think a problem arises in my final paragraph. It seems to me that I should not be able to say that $\mathcal{A}$ acts as a model for the infinite set of sentences. And even if that step is valid, it doesn't necessarily suggest that any fiber is infinite. And yet, the lemma the way is phrased, suggests that one may use the compactness theorem to prove it.
Are there any other avenues to approaching this problem via compactness?