Equivalent definitions of Poisson process

Define a Poisson process with parameter $\lambda$ is a counting process $(N(t))_{t\ge 0}$ such that:

(i) $N(0)=0$;

(ii) It has independent increment property;

(iii) $N(t+h)-N(t)$ has Poisson distribution with parameter $\lambda h$.

Now I want to prove this definition can lead to the other definition: Poisson process with parameter $\lambda$ is a renewal process in which the interarrival intervals has exponential distribution with parameter $\lambda$.

My proof: For all $n$, the event $X_n>t$ is the same as $N(t+S_{n-1})-N(S_{n-1})=0$ where $S_{n-1}=X_1+X_2+\cdots+X_{n-1}$. By (iii): $$P(X_n>t)=P(N(t+S_{n-1})-N(S_{n-1})=0)=e^{-\lambda t}$$

Thus $X_n$ has exponential distribution. By (ii), $X_n$'s are also independent.

Is my proof correct? If not, please help me with a correct one. Thank you very much.

• "Is my proof correct?" No, because it uses (iii) for the random time $t=S_{n-1}$ while (iii) is only assumed to hold for deterministic times $t$. – Did Jun 14 '15 at 20:07