TITLE: The limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
QUESTION [6 upvotes]: Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?

REPLY [13 votes]: Using the same "product-to-sum" strategy as @Did, we get
$${1\over 2}\prod_{k=2}^N\left({1-{1\over 2^k}}\right) 
\geq  {1\over 2}\left(1-\sum_{k=2}^N{1\over 2^k}\right) 
={1\over 4}+\left({1\over 2}\right)^{N+1}\geq   {1\over 4}.$$