Following is problem 1.1.9(b) in Problems in Mathematical Analysis II.
Show that if in a deleted neighborhood of zero the inequalities $f(x)\ge|x|^{\alpha}$, $\frac12<\alpha<1$, and $f(x)f(2x)\le|x|$ hold, then $\lim_{x\to0}f(x)=0$.
The given solution used $|x|^{\alpha}\le f(x)\le\frac{|x|}{|2x|^{\alpha}}$ and concludes that $\lim_{x\to0}f(x)=0$ (probably by the Squeeze Theorem). My question is, why do we need $\frac12<\alpha<1$? Is $0<\alpha<1$ not enough?