Suppose we have
start: with .
step: , . Find and .
So consider . We know that . Put . Then . Now how would we get ? Because taking inverse cosines, we get . So then . Thus . From here, can we get the limit?
No
You can show by induction that is increasing, is decreasing and
Starting from this you can show that has a limit since it is increasing and majored by . The same for .
Then you can show that and are adjacent since
Then you can show that and have the same limit l.
You can see that may be true but has infinite limit whereas converges towards