I have a question that goes like this: Suppose that F,K,L are all fields and L contains K which contains F. If L is a finite extension of F show that L is a finite extension of K and K is a finite extension of F.

One idea I had was letting A[i]B[j] be a bases of L over F and showing we can have A[i] as a bases of L over K and B[j] as a bases of K over F but I got stuck in the algebra and think I am making heavy work of it.

Is it enough to just say that as L is a finite extension over F both F and K must be field extensions and so as L contains them they must be finite dimensional? So K must be finite dimensional over F and L must be finite dimensional over F?

Thanks