We know the following inequality,
$$\left(1-\frac{1}{2}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{8}\right)\ldots\left(1-\frac{1}{2^{n}}\right)\geq\frac{1}{4}+\frac{1}{2^{n+1}}$$
We can find in these links the demonstration for induction of this result: [1], [2] and [3].
Obviously the result is still true if we change the right side by,
$$\frac{1}{4}\color{red}{-}\frac{1}{2^{n+1}}$$
Of course, I could use the previous inequality to complete this one, but I would like to know if it is possible to prove it directly by induction. It all comes down to proving this inequality,
$$\left(1-\frac{1}{2^{n+1}}\right)\left(\frac{1}{4}-\frac{1}{2^{n+1}}\right)\ge\frac{1}{4}-\frac{1}{2^{n+2}}$$
That it is not true, then, does that mean that it is not possible to show by induction? In this case, how would this result be demonstrated?