Termwise equivalent asymptotic scales need not preserve expansions

Even if ${\varphi }_n\sim {\psi }_n$ for every $n$, an expansion in the scale $({\varphi }_n)$ need not induce one in $({\psi }_n)$. A perturbation of ${\psi }_0$ that is smaller than ${\varphi }_0$ but larger than ${\varphi }_1$ supplies a counterexample.