I have a proof question regarding poisson processes.

I have a proof question regarding poisson processes. I know that you can prove that in a poisson process, the interarrival times follow an exponential distribution, and you can prove that the arrival times are uniform. I am wondering how you would prove this to be true for the discrete case, if the interarrival times are geometric, prove that the arrival times are discrete uniform, i.e. a discrete poisson process. Thanks!