Let's fix the issue of giving bounds on the infinite product. The pentagonal number theorem gives, after grouping pairs of consecutive terms with the same sign, an alternating series with terms that are decreasing in modulus. So for instance for $c:=\prod _ {n\ge1} (1-2^{-n})$ one has

$$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128} -\frac{1}{4096}-\frac{1}{ 32768 } < c < 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128}$$ that is $ 0.288787842 < c < 0.2890625, $ in any case larger than $1/4$.