Proof.
The lemma is trivial if the characteristic of K is zero; thus we may and do assume that the characteristic of K is p > 0.
Let K_2/K be a solution for A \to B. We will use induction on the inseparable degree [K_2 : K]_ i (Fields, Definition 9.14.7) of K_2/K. If [K_2 : K]_ i = 1, then K_2 is separable over K and we are done. If not, then there exists a subfield K_2/K_1/K such that K_2/K_1 is purely inseparable of degree p (Fields, Lemmas 9.14.6 and 9.14.5). By Lemma 15.116.5 there exists a separable extension K_3/K_1 which is a solution for A \subset B. Then [K_3 : K]_ i = [K_1 : K]_ i = [K_2 : K]_ i/p (Fields, Lemma 9.14.9) is smaller and we conclude by induction.
\square
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