= Solution
The <compactness theorem> states that a set of <first-order sentences> has a model if every finite subset has a model.
For each $n\geq1$, let $\chi_n$ assert that at least $n$ distinct elements exist. Consider $T=\{\phi\}\cup\{\chi_n:n\geq1\}$. A finite subset requires only some finite lower bound $N$ on the size. Choose an odd integer $m\geq N$; the given property of $\phi$ supplies a model of that size, satisfying the finite subset. By the <compactness theorem>, $T$ has a model. Its domain has at least $n$ elements for every $n$, so \b[$\phi$ has an infinite model]. This argument applies to every sentence with the finite-model-size property in (b), irrespective of its behavior on infinite structures.
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