= Solution
Again set $c=1$ and write
$$
X_A=x_C-x_A,\quad T_A=t_C-t_A,\qquad
X_B=x_C-x_B,\quad T_B=t_C-t_B.
$$
A boost of <velocity> $v$ gives
$$
X_A'=\gamma(X_A-vT_A),\qquad
X_B'=\gamma(X_B-vT_B).
$$
Define
$$
D(v)=(X_A-vT_A)^2-(X_B-vT_B)^2.
$$
The hypothesis says $D(0)<0$. Put $\xi=x_B-x_A$ and $\tau=t_B-t_A$. Causality gives $\tau\geq|\xi|$, while $T_B\geq|X_B|$, and $T_A=T_B+\tau$, $X_A=X_B+\xi$. Therefore
$$
T_A^2+X_A^2-(T_B^2+X_B^2)
=2T_B\tau+2X_B\xi+\tau^2+\xi^2>0.
$$
But
$$
D(1)+D(-1)
=2\{T_A^2+X_A^2-(T_B^2+X_B^2)\}>0.
$$
Hence at least one of $D(1),D(-1)$ is positive. By continuity, $D(v)>0$ for some physical <velocity> $|v|<1$ sufficiently close to that endpoint. In the corresponding inertial frame,
$$
|x_C'-x_A'|>|x_C'-x_B'|,
$$
as required.
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