Union of a strictly increasing sequence of closed theories in a finite language has an infinite model 
The union of a strictly increasing sequence of closed theories in a finite language has an infinite model.

This is an exercise in the book by Chang-Keisler. I don't have any idea to prove it. I just know that the union of a strictly increasing sequence of closed theories is a consistent closed theory which is not finitely axiomatizable, and also that the model of the union of consistent increasing theories is also the model of each theory.
If the union was complete the problem would be solved.
 A: I don't want to give away the whole problem, but here are a couple hints to get started:


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*First, show that if $\Sigma$ is a finite language then for each $n$ there are only finitely many $\Sigma$-structures of cardinality $\le n$ (up to isomorphism). In fact, more is true: every finite $\Sigma$-structure is characterized up to isomorphism by a single $\Sigma$-sentence. This isn't needed here, though.

*Next, remember that by the compactness theorem, if a theory has arbitrarily large finite models then it has infinite models.

*So suppose $(T_i)_{i\in\mathbb{N}}$ is an increasing sequence of  closed theories such that $T:=\bigcup T_i$ has no infinite model. Then some $T_i$ satisfies "there are at most $n$ elements in the domain" for some $n$; in particular, by the first bulletpoint there is some finite set of structures $\{\mathcal{M}_1,...,\mathcal{M}_k\}$ such that every model of $T_i$ is isomorphic to one of the $\mathcal{M}_j$s.

*Now, suppose $S_2$ is a closed theory properly containing $S_1$. What can you say about the set of models of $S_2$ versus the set of models of $S_1$? Looking back to the previous bulletpoint, what does this say about any increasing sequence $(T_i)_{i\in\mathbb{N}}$ of closed theories whose union has no infinite models?
