= Solution
For a massive particle, the <proper time> is determined by the <Minkowski interval> along its <worldline>:
$$
-c^2d\tau^2=-c^2dt^2+d\mathbf x\cdot d\mathbf x
=-c^2dt^2(1-|\mathbf u|^2/c^2).
$$
Choose future-directed time increments. With the <Lorentz factor> $\gamma_u=(1-|\mathbf u|^2/c^2)^{-1/2}$,
$$
\boxed{d\tau=dt/\gamma_u,\qquad U^\alpha=\frac{dx^\alpha}{d\tau}=\gamma_u(c,\mathbf u).}
$$
We use $U^\alpha$ for the source's $u^\alpha$ to distinguish <four-velocity> from <velocity> $\mathbf u$. Since <proper time> is invariant and $dx^\alpha$ is a <four-vector>, $U^\alpha$ is a <four-vector>, with $U_\alpha U^\alpha=-c^2$. The rest-frame result $U^\alpha=(c,\mathbf0)$ follows immediately.
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