Hi,
we can prove a) by induction:
base case: .... ....
inductive step: suppose and let's prove the statement holds for :
so the inductive step is completed and a) holds.
As for b), from a) we see that that the sequence is bounded from above by .
is the sum of first terms of a geometric progression, converges to 4/3.
So since is increasing, it is convergent.