In a Poisson Process with parameter $\lambda$, if $X_n$ is the time of the rth arrival.
Prove:
$$\sum^\infty_{n=1}fx_n(x)=\lambda\quad\text{for all}\ x\gt0$$
I’m new to Statistics and the Poisson Process in general. I know that $N(0) = 0$, that the number of arrivals in any interval $a\gt0$ has a Poisson$(\lambda a)$ distribution and that $N(t)$ increments are independent but I don’t see how any of that helps me to prove/show the given equation. I would appreciate any and all help.
$$\sum\limits^\infty_{n=1}f_{X_n}(x) \\= \sum\limits^\infty_{n=1} \dfrac{\lambda^{n-1} t^{n-1}e^{-\lambda t}}{(n-1)!}\lambda \\= e^{-\lambda t}\lambda\sum\limits^\infty_{n=1} \dfrac{(\lambda t)^{n-1} }{(n-1)!} \\= e^{-\lambda t}\lambda\sum\limits^\infty_{m=0} \dfrac{(\lambda t)^{m} }{m!} \\= e^{-\lambda t}\lambda e^{\lambda t} \\ = \lambda.$$