Intersection of infinite sets that contain each other

If each A_i is a set containing infinite elements, and A_1 contains A_2 contains A_3 contains ... on and on, then is the intersection of all these sets infinite?

I think the intersection of all the sets is infinite cause cause no matter how far you go down the A_i's that particular A_i will contain infinite elements and the intersection of all the sets before it, and including it, will be equal to A_i. So is my reasoning correct? Because maybe maybe as i approaches infinity the sets somehow get smaller?