Using calculus, prove (for ) that is monotone (increasing) and bounded above. Find its limit.
Expand sin(x) as x + O(x); then the limit of becomes obvious. To prove monotonicity and boundedness, expand as . This series satisfies the conditions to the alternating series test. From the info on that page, . Also, . You can show that , so .
Another way to show monotonicity is to consider and show that its derivative is eventually positive (also using series).