= Solution
Let $(M_i)_{i\in I}$ be nonempty <first-order structures> in one language and let $U$ be an <ultrafilter> on $I$. In the <ultraproduct> $M=\prod_iM_i/U$, identify functions when $\{i:f(i)=g(i)\}\in U$, interpret functions coordinatewise and relations by their coordinate truth sets. The <Łoś theorem> states that for every <first-order formula> $\varphi$,
$$
\boxed{M\models\varphi([f_1],\ldots,[f_k])\ \Longleftrightarrow\ \{i:M_i\models\varphi(f_1(i),\ldots,f_k(i))\}\in U.}
$$
All interpretations are independent of representatives: changing finitely many arguments on $U$-small sets changes truth only on their finite union, also $U$-small.
Prove the theorem by <structural induction> on formulas. Terms evaluate coordinatewise by induction on terms. This proves the atomic cases of <logical equality> and relation symbols. For conjunction, intersect the two truth sets and use the filter laws. For negation, the truth set is complemented, and an <ultrafilter> contains exactly one of a set and its complement. These cases supply the other Boolean connectives too.
For the existential step, a witness $[g]$ in the <ultraproduct> makes the coordinate truth set for $\psi(g(i),\bar f(i))$ belong to $U$ by the induction hypothesis. That set is contained in the coordinate truth set of $\exists x\,\psi(x,\bar f(i))$, so the latter is in $U$. Conversely, if that existential truth set $J$ belongs to $U$, choose a witness $g(i)$ in $M_i$ for each $i\in J$, and any element of $M_i$ elsewhere. The <axiom of choice> supplies this function. The induction hypothesis makes $[g]$ a witness in the <ultraproduct>. Universal quantification follows by negation. This completes the proof.
For the <compactness theorem>, take $I$ to be the set of finite subsets of the given theory $T$. For each $s\in I$, choose $M_s\models s$. For $\varphi\in T$, the cone $C_\varphi=\{s:\varphi\in s\}$ is nonempty, and any finite intersection of such cones contains their union as an index. Extend the filter they generate to an <ultrafilter> $U$. By the proved <Łoś theorem>, the <ultraproduct> satisfies every $\varphi\in T$, because its coordinate truth set contains $C_\varphi\in U$. Thus $\boxed{\prod_{s\in I}M_s/U\models T}$. This proves compactness without assuming it in the ultraproduct construction.
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