Let $L$ be a language and $T$ and $L$ theory. Suppose that for any $M\models{T}$, we have $M\subseteq{\bigcup{C_{n}}}$, where each $C_{n}\models{T_{\forall}}$ is finite. I want to show that for some finite subset $X$ of $M$, there is a finite $A\subseteq{M}$ such that $X$ is a subset of $A$.
This is related to the question: Definition of Local Finiteness
I have tried to follow a similar proof, but here I run in to a problem; namely the substructure generated by $X$ maybe infinite inside of $M$ but (is necessarily) finite inside of $\bigcup{C_{n}}$. I'm not sure how to use the given conditions to obtain that this does not occur.
Edit 1) As per the comments below: A is a substructure of $M$ and the $C_{n}$ form a chain.
Since $X$ is finite, there is some $n$ such that $X\subseteq C_n$ (note that this also works when the $C_n$ form a directed system instead of a chain). Now let $A$ be the substructure generated by $X$. Since $C_n$ and $M$ are structures containing $X$, we have $A\subseteq C_n$ (so $A$ is finite) and $A\subseteq M$.
You wrote "the substructure generated by $X$ may be infinite inside of $M$ but (is necessarily) finite inside of $\bigcup C_n$". I'm not sure I understand the issue here. But if it's not already clear to you, you should check that if $X$ is a subset of $A$, which is a substructure of $B$, then the substructure of $A$ generated by $X$ is equal to the substructure of $B$ generated by $X$.