Certainly if no element generates an inseparable extension then K/F is separable. That is, every element K/F is separable (of course assuming the extension is algebraic. This is not true if you use some black magic transcendental field extensions.).

It's essentially just by construction. To use Lang's terminology, separable extensions form a distinguished class, so they're stable under lifting (which is precisely what you're doing here.

http://books.google.com/books?id=FJmiSW1KRBAC&lpg=PP1&ots=k1ecm3FdbZ&dq=lang%20algebra&pg=PA242#v=onepage&q=&f=false

Edit: My mistake, I misread your post. Here's the correct answer.

http://books.google.com/books?id=FJmiSW1KRBAC&lpg=PP1&ots=k1ecm3FdbZ&dq=lang%20algebra&pg=PA251#v=onepage&q=&f=false

Proposition 6.11