= Solution
The <compactness theorem> states that a set $T$ of first-order sentences has a model if and only if every finite subset of $T$ has a model.
The forward implication follows by using the same model for every finite subset. Conversely, suppose every finite subset of $T$ has a model. If $T$ were inconsistent, a formal derivation of $\bot$ from $T$ would use only finitely many assumptions, say those in $T_0\subseteq T$. Then $T_0$ would be inconsistent. By part (b), $T_0$ would have no model, contrary to the hypothesis. Thus $T$ is consistent, and part (b) gives a model of $T$.
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