= Solution
Let $[L:F]=n<\infty$. The intermediate <field> $K$ is an $F$-linear <linear subspace> of $L$, so the finite-dimensional <linear subspace> theorem gives $[K:F]\leq n$. To handle the other <field extension>, take an $F$-<basis> $v_1,\ldots,v_n$ of $L$. Every $\ell\in L$ is a sum $\sum f_iv_i$ with $f_i\in F\subseteq K$, so the same list spans $L$ as a $K$-<vector space>. It need not remain linearly independent, but spanning suffices to give $[L:K]\leq n$. Hence \b[both intermediate extensions are finite].
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