 Solution

A rate-$\lambda$ Poisson process is a counting process starting at zero, with right-continuous nondecreasing integer-valued paths, independent stationary increments, and $N_t-N_s\sim \mathrm{Pois}(\lambda (t-s))$ for $0\le s<t$. Its jumps have size one almost surely.

For independent Poisson processes $N,M$, increments of $N+M$ on disjoint intervals are independent, because all the corresponding increments of the two processes are jointly independent. On an interval of length $t$, multiplying their probability generating functions givesThus the increment is Poisson with mean $(\lambda +\mu )t$. The path requirements hold, and independent processes almost surely have no simultaneous jumps. Therefore $N+M$ is a Poisson process of rate $\lambda +\mu$.