Could someone help with the following exercise. I am new to model theory and would appreciate any type of help.Thanks in advance!
Claim: Let $T$ be a $\forall \exists$- theory, $(A_i)_{ i \in \omega}$ a family of $\tau$-structures such that $A_i \models T$ and $(A_i)_{i \in \omega}$ form a chain. Show that $A= \bigcup_{i \in \omega} A_i$ defines naturally a $\tau$-structure $A$ that is a model of $T$.