LTL Until Tautology

Question

Today I had to prove some tautologies as an exercise, one of them was:

(Ga)->(Fb) equivalent aU(b v ¬a)

It is clear that I have to prove the implication in both ways. After putting into negation normal form i get that:

F(¬a v b) -> aU(b v ¬a) and viceversa.

I managed to prove the opposite implication, since aU(b v ¬a) implies that at some point we will either see b or ¬a, but I couldn't get around proving the other direction. I am sure it is because we have a and ¬a on the opposing sites of U, but the reasoning is not clear to me.

Can someone clear this up for me?