 Strong law for Poisson arrival times

For a homogeneous Poisson process of rate $\lambda >0$, $N(t)/t\to \lambda$ almost surely: unit-time increments are independent Poisson variables and the strong law of large numbers applies, then monotonicity fills the intervals between integers. At its $n$th point $Y_n$, simplicity gives $N(Y_n)=n$. Consequently $Y_n/n\to 1/\lambda$. Rate and mean interarrival time are reciprocals.