its seems like a fool proof what else do i need to prove here?

case1: An<An+1
case2: An>An+1
case3: An=An+1

if case1 is true:
then MAX will pick the member with the biggest n which is An^2
and you showed how to prove that if An->0 then An^2->0

if case 2 is true:
it will pick An and by definition An goes to 0

if case3:
then no matter what it will pick because its the same.
and all the members are equaled to An
so they all go to 0.
so it lookes pretty much complete
what do i miss??