I'm stuck on this too! Were you able to explain why a4, a8, a16, a32, ... is increasing? I can do it geometrically, since obviously a polygon that is inscribed in a circle with a diameter of 1 with 4 sides would have a smaller perimeter than one with 8 and in turn the perimeter of a polygon with 8 sides would be smaller than the one of a polygon with 16 sides. But mathematically?
And I can't for the life of me figure out a proof for the product rule, ie prove that if an tends to zero and bn tends to zero then anbn tends to zero.