Virtually amenable implies amenable?

Hello!

I've just recently been introduced to the many equivalent definitions of amenability (including Folner's condition).

I have also read that if a (discrete - ignoring topology for now) group G is virtually amenable (has a subgroup H of finite index where H is amenable), this implies that G is amenable.

I am looking for an elementary proof of this, and in particular, one which employs Folner's condition as the condition for amenability.

I have tried working on the problem in the case that H has index 2 in G (that is, G = H union gH for some g in G so that the cosets H and gH are disjoint) but to no avail.

Or, is there some other well-known elementary proof? Having trouble tracking anything down.

Thanks so much!