Prove that $\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $ with $R > 0$

Question

Sorry to bother you with silly question, but I can't figure out how to prove:

$$\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $$

with $R > 0$. I tried different ways but that didn't lead me anywhere. Can you give me some hints, or help me prove it ? Thanks

Answer 1

Multiplying throughout by $4e^R$ to get

$$2(e^{2R})-2\ge e^{2R}\iff e^{2R}\ge2\iff2R\ge\ln2$$

Answer 2

It isn't true. Note that for $R=0$ the left side is $0$ and the right side $1/4$. Since both sides are continuous, that inequality persists for some $R > 0$.