= Solution
Replace the applied field in the doublet response by the <Weiss molecular field> $B_E=\mu_0\lambda M$. At zero applied field,
$$
M=n\mu_B\tanh\left(\frac{\mu_0\lambda\mu_BM}{k_BT}\right).
$$
The equation is odd in $M$, so the two ordered orientations have equal magnitude. With $m=|M|/(n\mu_B)$, define
$$
\boxed{T_C=\frac{\mu_0\lambda n\mu_B^2}{k_B},\qquad t=\frac{T}{T_C}.}
$$
The <mean-field self-consistency equation> becomes
$$
\boxed{m=\tanh(m/t).}
$$
Plot $y=m$ and $y=\tanh(m/t)$. The second has slope $1/t$ at zero and is concave for positive $m$. For $0<t<1$ it starts above the straight line but is below it by $m=1$, giving one positive intersection. For $t\geq1$, $\tanh(m/t)<m$ for every $m>0$. Hence
$$
\boxed{m>0\text{ exists for }0<T<T_C;\qquad m=0\text{ for }T\geq T_C.}
$$
At $T=0$ the limiting solution is $m=1$. The onset of <spontaneous magnetization> is the mean-field <Curie temperature>.
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