# A inequality about infinite product $\prod\limits_{i=1}^\infty(1-\frac1{2^i})$

Show that $$\prod_{i=1}^\infty(1-\frac1{2^i})>0.288$$

To get the required bound, you need to modify my argument from the other answer: $$\prod_{k=1}^\infty \left({1-{1\over 2^k}}\right) \geq \prod_{k=1}^5\left({1-{1\over 2^k}}\right) \left(1-\sum_{k=6}^\infty{1\over 2^k}\right) ={315\over 1024}\cdot {15\over 16}={4725\over 16384}\approx .28839.$$