Consider the following statement: "Let $L/K$ be an algebraic field extension. Then the maximal separable sub-extension is finite."
Here is what seems to be a proof: "Let $M/K$, $K \subset M \subset L$ be a finite separable subextension of maximal degree. Let $x \in L$ be separable over $K$. So it will be separable over $M$. Consider the field tower $K \subset M \subset M(x)$. Each step of this tower is separable and finite and so $M(x) / K$ is finite and separable. But then $x \in M$ by the maximality assumption. This shows that $M/K$ is in fact the maximal separable sub-extension."
Could you confirm the validity of the above proof? Thanks.