= Solution
For $n\geq1$, put $r_n=\phi_n/\phi_{n-1}$. Since $(\phi_n)$ is an <asymptotic sequence>, $r_n\to0$. We can rewrite
$$
\psi_n=\frac{\phi_n}{1+r_n},
$$
so
$$
\frac{\psi_n}{\phi_n}=\frac1{1+r_n}\longrightarrow1.
$$
Together with $\psi_0=\phi_0$, this proves
$$
\boxed{\psi_n\sim\phi_n\quad\text{for every }n\geq0.}
$$
Moreover,
$$
\frac{\psi_{n+1}}{\psi_n}
=\frac{\psi_{n+1}}{\phi_{n+1}}
\frac{\phi_{n+1}}{\phi_n}
\frac{\phi_n}{\psi_n}
\longrightarrow1\cdot0\cdot1=0.
$$
\b[Hence $(\psi_n)$ is an asymptotic sequence. This is the <harmonic-mean refinement of an asymptotic sequence>.]
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