I have a question when reading Essentials of Stochastic Process by Richard Durrett, 2.2.1 Constructing the Poisson Process. It says, Let $\tau_1,\tau_2,\dotsc$ be independent exponential$(\lambda)$ random variables. Let $T_n=\tau_1+\dotsc+\tau_n$ for $n \geq 1, T_0=0$, and define $N(s) = max \{ n:T_n \leq s \}$.
To show that $N(t)$ is a Poisson process, the book first shows $N(s)$ has a Poisson distribution with mean $\lambda s$. To show that $N(t)$ has independent increments, the book proves this Lemma: $N(t+s) -N(s), t\geq 0$ is a rate $\lambda$ Poisson process and independent of $N(r), 0 \leq r \leq s.$ Here it has this statement that I can't wrap my head around, "... the interarrival times after s are independent exponential$(\lambda)$, and hence that $N(t+s) -N(s), t\geq 0$ is a Poisson process." screenshot of the page
It confuses me because the whole section is trying to show that $N(t)$ is a Poisson process via the definition, (i) $N(0) = 0$, (ii)$N(t+s) -N(s)=Poisson(\lambda t)$, and (iii) $N(t)$ has independent increments. But when proving independent increments, it just proves that $N(t+s) -N(s), t\geq 0$ is a Poisson process. It seems that "the interarrival times are independent exponential$(\lambda)$" is sufficient for $N(t+s) -N(s), t\geq 0$ to be a Poisson process, but then by the same logic, isn't $N(t)$ automatically a Poisson process?
If anyone can explain to me why is $N(t+s) -N(s), t\geq 0$ a Poisson process, or why are the increments independent, that would be much appreciated.
