Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable.
Using the Element Primitive Theorem, we know that $M=K(a)$ and $M=K(b)$, hence $MN=K(a)K(b)=K(a,b)$ therefore $MN$ is separable since $a$ and $b$ are separable.
My proof is correct?
Nothing wrong if you are allowed to use the primitive element theorem. However, note that the result is true even if the extensions are infinite. Specifically, given any field extension $K / F$ with separable subextensions $M / F$ and $N / F$, $MN / F$ is separable. To prove this, you just take any element in $MN$ and consider how it was generated, which needs only finitely many elements in $M,N$. By adjoining these to $F$ we get a chain of separable extensions which must be separable.