Let $\alpha\in [0,1)$. I would like to prove that there exists a constant $c>0$ such that
$$\prod_{i=0}^{+\infty}(1-\frac{1}{2}\alpha^i)>c>0$$.
I tried writing it in the exponential form and after using the Taylor expansion, in the following way: \begin{align} \prod_{i=0}^{+\infty}(1-\frac{1}{2}\alpha^i)= \prod_{i=0}^{+\infty}e^{\log(1-\frac{1}{2}\alpha^i)}=\prod_{i=0}^{+\infty} e^{-\frac{1}{2}\alpha^i-\frac{1}{8}\alpha^{2i}-\frac{1}{24}\alpha^{3i}-\cdots} \end{align} But I couldn't succeed. Has someone any idea?
Thank you very much!