Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. Let $S_k$ be the sequence of times at which the $k$th event occurs, ie. the renewal times. $S_k = X_1 + \ldots + X_k$ so that $S_k$ has a negative binomial distribution with parameters $k$ and $p$.
Now consider the joint distribution of two consecutive renewal times.
How do you show that: $P(S_k \leq n ; S_{k+1} = n+j) = P(S_{k+1} = n+1)(1-p)^{j-1}$ for $n \geq k$?