i don't think this is true, without more information on F. for one can devise algebraic, but not separable, extensions of F and then G, in which case L is NOT separable over F.
perhaps F is a field of characteristic 0?
Prove that if L/G and G/F are separable algebraic extensions (not necessarily finite), then L/F is also separable.
I could do it for the finite case, but I'm not sure what to do here?
Ohh I'm sorry, that should have said, "if L/G and G/F are *separable* algebraic extensions".
And we already proved in class that if L/G and G/F are both algebraic extensions (not necessarily finite) then L/F is an algebraic extension. So I just need to show L/F is separable too.