Perhaps a little clearer: .
a) Prove first that (Hint: For odd
and etc. (induction, of course). For even n do something simmilar)
b) Next...induction, again and this time let us do the one with even indexes: we want to prove that
.. and etc., and something simmilar to show
In fact the above's enough to show that both sequences converge. Now you may want to show they both converge and to the same limit...