Consider the following product:
$$ \prod_{i=1..n} {\left(1 - {1 \over 2^i}\right)} $$
A numeric calculation, up to $n=20$, gives $0.288788370496567$. But how can I calculate its limit when $n$ goes to infinity?
Alternatively, how can I prove that, for every $n$, the product is larger than $0.25$ (or some larger constant)?
Hint:
$$\prod_{k=1}^n 1-{1\over 2^k} = {1\over2}\left(\prod_{k=1}^{n-1} 1-{1\over2} \left({1\over2}\right)^k\right) = {1\over2}\left ( \prod_{k=1}^{n-1}1-2^{-k-1} \right) \implies$$ $$\prod_{k=1}^\infty 1-{1\over 2^k} = {1\over2}\left (\prod_{k=1}^{\infty}1-2^{-k-1} \right)$$