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.