Solution
= Solution
Successive radial wave crests travel the same fixed comoving distance, so
$$
\int_{t_e}^{t_0}\frac{dt}{a(t)}
=\int_{t_e+\delta t_e}^{t_0+\delta t_0}\frac{dt}{a(t)}.
$$
To first order, $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Since wavelength is proportional to the period,
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_e}=\frac{\delta t_0}{\delta t_e}
=\frac{a(t_0)}{a(t_e)}.}
$$