I'm finding it difficult to solve the first order differential equation in the proof to obtain the conclusion. I've solved 1st order ODE's before and not had problems and I feel really silly for not being able to do this. I know this is separable, but I can't get the solution in the required form.
$$ \frac{{\rm d}W}{{\rm d}t} = -pW $$
On
This is probably an improvement of the answer by user caverac: to avoid assuming that $W>0$ (which is required in that answer) multiply the equation $\frac {dW} {dt} +pW=0$ by the integrating factor $e^{\int_{x_0}^{x} p(t)\, dt}$ to get $\frac d {dt} (We^{\int_{x_0}^{x} p(t)\, dt})=0$. Hence $We^{\int_{x_0}^{x} p(t)\, dt}$ is a constant $c$ and $W=ce^{-\int_{x_0}^{x} p(t)\, dt}$. This avoids dividing by $W$ and taking logarithm of $W$.
Write the equation as
$$ \frac{{\rm d}W}{W} = -p {\rm d}t $$
Integrate both sides
$$ \ln W = - \int_{x_0}^x p(t){\rm d}t + \tilde{C} $$
take the exponential
$$ W = C\exp\left(- \int_{x_0}^x p(t){\rm d}t\right) $$
where $C = e^{-\tilde{C}} : {\rm const}$