So I was toiling away trying to show that a sequence of random variables converges to a particular distribution by showing that the characteristic functions of the limit and the target are equal, however I found myself stuck when trying to bridge the following gap in my computations,
\begin{align} \lim_{n\rightarrow\infty}\prod_{k=1}^{n}\cos{\left(\frac{t}{2^{k}}\right)} \stackrel{?}{=} \frac{\sin{(t)}}{t}. \end{align}
So if I can close that equality in some way (by some proof) I am done, but this seems as a harder task than I expected, or even what the problem is designed for. I have search the forum for clues, but there are only similar infinite products of rational functions, not trigonometric functions.
Is the equality true? How do we show that?
