Nice proof of the Primitive Element Theorem for infinite base fields (though it does rely on having done stuff in a different order to Yoshida). We will assume two facts, and that $K$ is any infinite field.

Fact 1: Every separable extension has only finitely many subextensions.

Fact 2: No finite-dimensional vector space over an infinite field is a union of finitely many of its proper subspaces.

Theorem (Primitive Element Theorem): If F/K is finite and separable then it has a primitive element ($\theta$ such that $F=K(\theta)$).

Proof: Let $X = F \backslash (\bigcup_{K\leq M < F} M)$. By fact 1, this is a finite union and so by fact 2, $X$ is nonempty. Hence any element of $X$ is a primitive element.