Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 1 / a / Solution

Expand the language by constants for every element of $M$ and a new tuple $c$. Let $\mathrm{ElDiag}(M)$ be the elementary diagram of a structure associated with $M$. Every finite subset ofis realized in the expansion of $M$, because the finitely many formulas from $p$ are simultaneously satisfiable in $M$. The compactness theorem gives a model $N^{\prime }$ of the whole set. The interpretations of the named constants give an elementary embedding $M\to N^{\prime }$, and the interpretation of $c$ realizes $p$. Identifying $M$ with its image produces an elementary extension $M⪯N$ realizing $p$. This is the finitely satisfiable type is realized in an elementary extension argument.