= Solution
Take the ideal <electric dipole> limit with a fixed smooth <electric dipole moment> $\mathbf p(t)$, so that $\mathbf r_0(t)=\mathbf p(t)/(2q)$; its required time <derivatives> are bounded on the time interval under consideration. Fix $r>0$ and put $\mathbf n=\mathbf r/r$, $t_0=t-r/c$. For $s=+1$ or $-1$, the source position is $s\mathbf p(t)/(2q)$. A <Taylor expansion> of the distance gives
$$
R_s(t_r^s)=\left|\mathbf r-\frac{s\mathbf p(t_r^s)}{2q}\right|
=r-\frac{s\mathbf n\cdot\mathbf p(t_r^s)}{2q}+O(q^{-2}).
$$
Substitute this into the defining equation for <retarded time>:
$$
t_r^s=t_0+\frac{s\mathbf n\cdot\mathbf p(t_r^s)}{2qc}+O(q^{-2}).
$$
Boundedness first implies $t_r^s-t_0=O(q^{-1})$. The <Taylor expansion> $\mathbf p(t_r^s)=\mathbf p(t_0)+O(q^{-1})$ then changes the displayed correction only by $O(q^{-2})$. Hence
$$
\boxed{t_r^+=t_0+\frac{\mathbf p(t_0)\cdot\mathbf r}{2qrc}+O(q^{-2}),\qquad
t_r^-=t_0-\frac{\mathbf p(t_0)\cdot\mathbf r}{2qrc}+O(q^{-2}).}
$$
The nearer positive displacement reduces the propagation distance and consequently increases the <retarded time>. The <Big O notation> here is for $q\to\infty$ at a fixed observation event away from the origin. Keeping a fixed smooth $\mathbf p$ is essential: bounded amplitude alone, with arbitrarily increasing time frequency as $q$ varies, would not justify this <Taylor expansion>.
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