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First,For $n\ge 1$,Thus $({\chi }_n)$ is an asymptotic sequence.

Termwise asymptotic equivalence fails for every $n\ge 1$, becauseFor example, as $x\to 0$, the positive scale ${\varphi }_n(x)=|x{|}^n$ gives ${\chi }_n(x)=|x{|}^{n-1/2}$ for $n\ge 1$. This illustrates the geometric-mean refinement of an asymptotic sequence.