Finding $\lim\limits_{n \to \infty}(\sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n))$ Problem: Find $\lim_{n \to \infty}(\sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n))$.
My idea: I suppose it to be $0$, but how might I go about proving this?

$$\sin(k)\sin(k+1)=\frac{1}{2} [\cos(1)-\cos(2k+1)]$$
Now use the fact that $.5 < \cos(1) < .6$ to conclude that 
$$\left| \sin(k) \sin(k+1)\right| < 0.8$$
and 
$$\left| \sin(1)\cdot \sin(2) \cdot \ldots \cdot \sin(n) \right| \leq 0.8^{\frac{n}{2}}$$
