<s>[INST] Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$? Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
 [/INST] Recall either of Euler's two famous expressions for $\sin x$:
$$\sin x=x\prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),$$ 
or
$$\sin x=x\prod_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right).$$
Now let $x=\dfrac{\pi}{6}$.
Or else use the following formula of Viète
$$\frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}\cdots,$$
and multiply both sides by $\dfrac{\pi}{4}$. 
</s>