E/F finite extension F'/F any extension. EF' composite field. Prove [EF':F']<=[E:F]

Question

As the question says (we assume E and F' are contained in some field so that the composite EF' is well-defined). I've thought of trying to prove this using indiction by writing E=F(a1,...ar) but unfortunately I haven't really been able to get onto the right track and cant seem to quite get to the answer. Do you know of any hints/ways to solve this problem? Thank you for any comments or answers.

EDIT: I have now managed to solve the base case E=F(a).

Answer 1

lets assum that F'=F(x), then EF=EF(x)=E(x), the minimal polynomial of x over F (say f) is a polynomial in E[X], so the minimal polynomial of x over E must devise f, so [EF':E]<=[F':F].