Remark 1

The restriction on $g$ in prop. 7 is essential. For example, one has $\frac{1}{x} \ll 1 + \frac{1}{x}$ even though the derivatives of the two sides are equivalent; similarly $1 + \frac{1}{x} \sim 1 + \frac{1}{x^2}$, but $1/x^2 \gg 2/x^3$.