= Solution
For a radial <null geodesic> in the <FRW metric>, define $\chi(r)=\int_0^r(1-kr'^2)^{-1/2}\,dr'$. Two successive wave crests emitted and received at the same comoving positions obey
$$
\chi_e=c\int_{t_e}^{t_0}\frac{dt}{a(t)}
=c\int_{t_e+\Delta t_e}^{t_0+\Delta t_0}\frac{dt}{a(t)}.
$$
Subtracting and keeping the first order in the wave periods gives $\Delta t_0/a(t_0)=\Delta t_e/a(t_e)$. Thus the <cosmological time dilation> and the <cosmological redshift> have the same factor:
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_e}
=\frac{\nu_e}{\nu_0}
=\frac{\Delta t_0}{\Delta t_e}
=\frac{a(t_0)}{a(t_e)}=\frac1{a(t_e)}.}
$$
Here the emitter and observer are comoving; an additional <peculiar velocity> would supply a separate <kinematic redshift>. The derivation uses the <null geodesic> propagation of the crests rather than interpreting a finite cosmological distance as an ordinary <Doppler effect>.
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