 Solution

Choose the two inertial frames to have coincident origins at $t=t^{\prime }=0$, and let the origin of $S^{\prime }$ have $x=vt$. The Lorentz transformation isHere ${\gamma }_v$ is the Lorentz factor. Along the particle's trajectory, $dx/dt=u$, soDividing gives the relativistic velocity-addition formulaThe denominator is positive because $|uv|<c^2$. Direct algebra yieldssince both factors in the numerator are positive. Consequently $\boxed{|u^{\prime }|<c}$: a subluminal velocity remains subluminal in every such inertial frame.