= Solution
\b[<redshift>.] An incoming radial null ray satisfies $d\chi=-c\,dt/R(t)$. Consecutive wave crests follow the same comoving path, so the endpoint changes in the fixed path <integral> obey $dt_0/R_0=dt_e/R_e$. Comoving clocks read <cosmic time>; hence their measured periods and frequencies satisfy
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_e}=\frac{dt_0}{dt_e}=\frac{R_0}{R_e}}.
$$
The physical <wavelength> stretches with the <scale factor> during propagation. It is not necessary to assign a unique global inertial-frame recession <velocity> to a distant source.
\b[Radial coordinate and <cosmological look-back time>.] Differentiating $1+z=R_0/R(t)$ gives $dz/dt=-(1+z)H(z)$. Combining this with the null-ray <integral> yields
$$
\boxed{\chi(z)=\frac c{R_0}\int_0^z\frac{d\bar z}{H(\bar z)},\qquad
t_0-t_e=\int_0^z\frac{d\bar z}{(1+\bar z)H(\bar z)}}.
$$
The extra factor $1+z$ in the second denominator records the conversion from scale-factor change to elapsed <cosmic time>.
\b[<proper-motion distance>.] A small transverse displacement at emission has proper length $d\ell=R_eS(\chi)d\theta$. Define its source-frame transverse speed as $v=d\ell/dt_e$; <cosmological time dilation> makes $dt_0=(1+z)dt_e$. Therefore
$$
\dot\theta_0=\frac{v}{(1+z)R_eS(\chi)},\qquad
\boxed{d_M=\frac v{\dot\theta_0}=R_0S(\chi(z))}.
$$
Equivalently, $d_M=(1+z)d_A$, where the <angular diameter distance> is $d_A=R_eS(\chi)$. The observer-time versus source-time distinction is essential. This calculation uses the stated transverse speed convention and the homogeneous background, without additional peculiar radial <relativistic Doppler effect> or <gravitational lensing>.
\b[<redshift shell volume at the present epoch>.] The induced spatial <metric tensor> on the present slice gives $dV_0=R_0^3S^2(\chi)d\chi\,d\Omega$. Since $d\chi/dz=c/(R_0H)$,
$$
\boxed{dV_0=\frac{cR_0^2S^2(\chi(z))}{H(z)}\,dz\,d\Omega
=\frac{cd_M^2(z)}{H(z)}\,dz\,d\Omega}.
$$
This is the volume at the present epoch of the comoving shell intersected by the past <light cone>. Its proper volume at emission is smaller by $(1+z)^3$; a conserved population's proper <number density> is larger by the same factor, preserving its count.
\b[<Einstein-de Sitter cumulative number counts>.] In flat geometry $S(\chi)=\chi$. Inserting the stated <Hubble parameter> into the radial <integral> gives
$$
d_M(z)=\frac{2c}{H_0}\left[1-(1+z)^{-1/2}\right].
$$
The constant present-volume <number density> makes $dN=n_0dV_0$. With no members of the population before its formation epoch,
$$
\boxed{N=\frac{4c^3n_0}{H_0^3}\int_0^{z_f}
\frac{[1-(1+z)^{-1/2}]^2}{(1+z)^{3/2}}\,dz}.
$$
As an independent check, substitution $u=(1+z)^{-1/2}$ evaluates this to
$$
\boxed{N=\frac{8c^3n_0}{3H_0^3}\left[1-(1+z_f)^{-1/2}\right]^3
=\frac{n_0d_M^3(z_f)}3}.
$$
This is exactly the Euclidean <comoving volume> per unit <solid angle> in the flat model. It assumes a complete count with the specified fixed population; <luminosity> selection, mergers or subsequent number-density evolution would require additional factors.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2013/paper-1-eds-distance-counts.png]
{title=Original Einstein-de Sitter proper-motion and angular diameter distances, differential counts and cumulative fraction}
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