= Solution
For $N\ge2$, each adjacent pair of walkways has relative <rapidity> $\phi_N=\operatorname{artanh}(1/N)$. There are $N$ such steps between $P_0$ and $P_N$, so the <equal-rapidity chain of collinear Lorentz boosts> has total <rapidity> $N\phi_N$. Thus the actual <velocity> is
$$
\boxed{v_N=c\tanh\left(N\operatorname{artanh}\frac1N\right)
=c\frac{\left(\frac{N+1}{N-1}\right)^N-1}{\left(\frac{N+1}{N-1}\right)^N+1}.}
$$
It is strictly below the <speed of light> for every finite $N\ge2$. Adding the adjacent <velocities> as ordinary numbers ignores the <relativistic velocity-addition formula>.
For large $N$, the <Taylor series> of the inverse <hyperbolic tangent> gives
$$
N\operatorname{artanh}(1/N)=1+\frac1{3N^2}+O(N^{-4}).
$$
Consequently
$$
\boxed{v_N\sim c\tanh1=\frac{e^2-1}{e^2+1}\,c\approx0.761594156\,c.}
$$
More precisely, $v_N/c=\tanh1+\operatorname{sech}^2(1)/(3N^2)+O(N^{-4})$, so the limit is approached from above. The case $N=1$ would prescribe an adjacent relative <velocity> equal to the <speed of light>; it does not define an admissible walkway <inertial frame>.
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