Uniform convergence of product of cosines

Question

Let $$f_n(x)=\prod_{t=1}^{n}\cos\frac{x}{2^t}$$ It is well-known that $$\lim_{n\to\infty}f_n(x)=\frac{\sin x}{x}=\text{sinc}(x)$$ Is the sequence $f_n$ uniformly convergent to $\text{sinc}$ over compact sets; over the entire real line?

Answer 1

Hint. One may recall that $$ x-\frac{x^3}{6}\le \sin x \le x, \qquad x\ge0, $$ giving, for $n\ge1$, $$ x-\frac{x^3}{2^{2n}}\le 2^{n}\sin\left(\frac{x}{2^{n}}\right) \le x, \qquad x\ge0, $$ then one may combine this with the telescopic product $$ f_n(x)=\prod_{t=1}^{n}\cos\frac{x}{2^t}=\frac{\sin x}{2^{n}\sin\frac{x}{2^{n}}}, \qquad x\ne0, \, n\ge1. $$