Proving $(1-\frac{1}{2})(1-\frac{1}{4})\dotsm(1-\frac{1}{2^n}) \geq \frac{1}{4}+\frac{1}{2^{n+1}}$ using induction

Question

Prove using induction: $(1-\frac{1}{2})(1-\frac{1}{4})\dotsm(1-\frac{1}{2^n}) \geq \frac{1}{4}+\frac{1}{2^{n+1}}$.

I know how to prove such equalities but not really inequalities. I tried multiplying both sides by $1-\frac{1}{2^{n+1}}$ and rearrange in order to obtain the same expression as above but for $m=n+1$, but to no avail.

Answer 1

$({1\over4}+{1\over2^{n+1}})(1-{1\over2^{n+1}})={1\over4}+{3\over4}({1\over2^{n+1}})-{1\over2^{2n+2}}={1\over4}+{3\over2}({1\over2^{n+2}})-{1\over2^{2n+2}}=({1\over4}+{1\over2}({1\over2^{n+2}}))+({1\over2^{n+2}}-{1\over2^{2n+2}})>{1\over4}+{1\over2}({1\over2^{n+2}})$.

So your solution is correct, you just need to be more patient and do the calculations.