All this is much more than we need, but at least it tells us that odd subsequence is increasing while the even subsequence is decreasing. To know that they converge we prove that the subsequences are bounded. To do this we need to show that the odd terms are smaller than the even terms. But using pre-calculus, this is obvious. Even terms are smaller than , so the odd subsequence is increasing, bounded and so converges.
Odd terms are larger than .
So the even subsequence is decreasing, bounded and so converges. Just for fun, let N be the limit of the even terms. So the even terms converge to 1. Similarly the odd terms also converge to 1.