Well it's a two part question, with the idea being that in the end we're going to prove rank(AH) = rank(A)
Part 1 is what you did above. To prove that rank(AH)<=rank(A).
Part 2 is to prove rank (AHH^-1) <= rank (AH).
By proving the two statements, we'd show that rank(AH)=rank(A).
A hint that was given was to consider how we reduce to row echelon form.
Well guess this might work then then Put B = AH, and X= H^-1 in part 1
Then
rank(BX) <= rank (B) (like the logic used in part 1)
Put the values back in we get your part 2
Only thing now remains rank of (AH) <= min[rank A, rank H]
Which is easy if we consider AH as linear combination of column vectors of A OR equivalently linear combination of row vectors of H.
Hope it helps