Corollary

*If two real functions $f, g$, are comparable of order $k \geqslant 1$, then they are comparable of order $p$ for $0 \leqslant p \leqslant k$; further, the relation $f \ll g$ (resp. $f \sim cg$) implies $f^{(k)} \ll g^{(k)}$ (resp. $f^{(k)} \sim cg^{(k)}$) unless one of the derivatives $g^{(p)}$ ($0 \leqslant p \leqslant k - 1$) is equivalent to a constant $\neq 0$.

Indeed, since $f^{(k)}$ and $g^{(k)}$ have constant sign on an interval $[x_0, +\infty[$, it follows that $f^{(k-1)}$ and $g^{(k-1)}$ are monotone on this interval, so have constant sign on a neighbourhood of $+\infty$; further, prop. 7 of V, p. 232, shows that $f^{(k-1)}$ and $g^{(k-1)}$ are comparable, so the corollary follows from applying prop. 7 recursively.