Let K ⊆ E ⊆ F be fields. Prove that if F is algebraic over K, then F is algebraic over E and E is algebraic over K.

Here is what I am thinking:

Since F is algebraic over K, then F would be algebraic over E since E ⊆ F. Similarly, E would be algebraic over K since E ⊆ F and F is algebraic over K.

Does this make sense for a proof? If not, can you please help? Thanks in advance.