= Solution
Apply the preceding <temperature> profile at the envelope base. Introduce the dimensionless parameter
$$
q=\frac{17}{4}\frac{\mathcal RT_cR}{GM}.
$$
Then $R/r_b-1=q$, so the <geometrical thickness of a white-dwarf radiative envelope> is
$$
\boxed{r_b=\frac{R}{1+q},\qquad
\frac sR=\frac{q}{1+q}\simeq q,\qquad
s\simeq\frac{17}{4}\frac{\mathcal RT_cR^2}{GM}.}
$$
The smallness to be established is therefore $\mathcal RT_c\ll GM/R$, up to the numerical factor $17/4$.
That ordering follows from the cold, strongly electron-degenerate <white dwarf> model. Throughout the mass-bearing core, $kT_c\ll E_F$, where $E_F$ is the <Fermi energy>. For nonrelativistic <electron degeneracy pressure>,
$$
\frac{P_{\rm deg}}\rho=\frac{2E_F}{5\mu_em_H}.
$$
The <stellar hydrostatic equation> gives a characteristic core $P_{\rm deg}/\rho$ of order $GM/R$. Consequently
$$
\frac{\mathcal RT_c}{GM/R}\sim\frac{5\mu_e}{2\mu}\frac{kT_c}{E_F}\ll1
$$
for fixed composition with order-one molecular-weight ratios, and \b[$s\ll R$]. Equivalently, the matched base <mass density> is much lower than a characteristic core <mass density>, since $\rho_b/\rho_{\rm core}\sim[\mathcal RT_c/(GM/R)]^{3/2}$. The degeneracy ordering applies in the bulk core; at the matching point degeneracy is becoming marginal.
The inequalities $T_c\gg T_{\rm eff}$ and $kT_c\ll E_F$ express different facts. The first justifies ignoring surface-temperature terms, while the second makes the envelope geometrically thin. If the core were hot enough that $q$ were not small, the exact expression $s/R=q/(1+q)$ would not justify a thin-layer approximation.
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