Answer
See the detailed answer below.
Work Step by Step
We need to prove that $$f_-=\dfrac{f_0}{1+\dfrac{v_s}{v}}$$
which is the frequency measured when the source moves away from the observer.
Let's assume that the distance $d$ is the distance the wave has moved plus the distance the source has moved at a time of $t = nT$; where $t=0, T,2T,...nT$, $n$ is a positive integer number.
These distances are
$$\Delta x_{wave}=nvT $$
$$\Delta x_{source}=nv_sT $$
where $$\lambda_-=\dfrac{d}{n}=\dfrac{\Delta x_{wave}+\Delta x_{source}}{n}= vT+v_sT $$
$$\lambda_-=(v+v_s)T $$
The frequency detected by you when the source is moving away is then
$$f_-=\dfrac{v}{\lambda_-}=\dfrac{v}{(v+v_s)T}=\dfrac{vf_0}{v+v_s}$$
Therefore,
$$\boxed{f_- =\dfrac{ f_0}{1+(v_s/v)}}$$
