Prove that $\prod_{i=1}^\infty \left(1-\frac{1}{5^i}\right) >3/4$

Question

Prove that $$\prod_{i=1}^n \left(1-\frac{1}{5^i}\right) >\frac34.$$

Any hint?

Answer 1

Hint. Note that $$\prod_{i=1}^n \left(1-\frac{1}{5^i}\right)\geq 1-\sum_{i=1}^n\frac{1}{5^i}>1-\sum_{i=1}^{\infty}\frac{1}{5^i}.$$ You may show the first inequality by induction.