= Solution
Use the inverse <Lorentz transformation>
$$
x=\gamma_v(x'+vt'),\qquad
t=\gamma_v(t'+vx'/c^2).
$$
Along the particle's worldline $dx'=u'\,dt'$, so the <relativistic velocity-addition formula> is
$$
\boxed{u=\frac{u'+v}{1+u'v/c^2}.}
$$
For subluminal collinear <velocities>, put $u/c=\tanh\varphi_u$, $u'/c=\tanh\varphi_{u'}$ and $v/c=\tanh\varphi_v$. The <hyperbolic tangent> addition formula, obtained by dividing the supplied sine/cosine addition formulas, gives
$$
\tanh\varphi_u=\tanh(\varphi_{u'}+\varphi_v).
$$
Because <hyperbolic tangent> is <injective> on the real line,
$$
\boxed{\varphi_u=\varphi_{u'}+\varphi_v.}
$$
Thus <rapidity> is additive even though <velocity> is not.
Each positive increment in the instantaneous rest frame adds the same <rapidity> $\alpha=\operatorname{artanh}(1/2)=\tfrac12\log3$. Starting from zero <rapidity>, <iterated collinear boosts with equal rapidity> give
$$
\boxed{u_n=c\tanh(n\alpha)
=c\frac{e^{2n\alpha}-1}{e^{2n\alpha}+1}
=c\frac{3^n-1}{3^n+1}.}
$$
Every finite number of increments leaves $u_n<c$; the <speed of light> is approached as $n\to\infty$.
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