Missing argument in the proof of the Levy-Khintchine representation .

In one proof of the Levy Khintchine representation of the Laplace exponent of subordinators, the following argument is used:

"Assume $f_n : [0,\infty) \to [0,\infty)$ is a sequence of non-increasing functions with

$$\int_{0}^{\infty} e^{-\lambda x} f_n(x) dx \; \text{ converges towards a finite value as } \; n \to \infty \text{ for each }\; \lambda \geq 0,$$

then necessarily the limit has the form

$$\int_{0}^{\infty} e^{-\lambda x} (\alpha \delta_0(x) + f(x) dx) = \alpha + \int_{0}^{\infty} e^{-\lambda x} f(x) dx$$

for some $\alpha\geq 0$ and some non-increasing function $f$."

How to prove this statement ?

It is quite unusual to be able to say anything about the "densities" under such a weak convergence statement. But the non-increasing assumption is crucial here.