= Solution
Recall that positivity and asymptoticity mean
$$
\frac{\phi_{j+1}}{\phi_j}\to0,\qquad
\frac{\psi_{j+1}}{\psi_j}\to0
$$
for each fixed $j$. For $0\leq k\leq n$, compare the $k$th term of
$\chi_{n+1}$ with the $k$th term of $\chi_n$:
$$
\frac{\phi_k\psi_{n+1-k}}{\phi_k\psi_{n-k}}
=\frac{\psi_{n+1-k}}{\psi_{n-k}}\to0.
$$
For the remaining term, compare with the final term of $\chi_n$:
$$
\frac{\phi_{n+1}\psi_0}{\phi_n\psi_0}
=\frac{\phi_{n+1}}{\phi_n}\to0.
$$
Since $\chi_n$ is positive and dominates every comparison term, summing these finitely many bounds gives
$$
\boxed{\frac{\chi_{n+1}(x)}{\chi_n(x)}\to0}.
$$
\b[Thus $(\chi_n)$ is an asymptotic sequence.]
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