My book outlines a procedure but a preliminary part of it is unclear to me.
Let X be the number of occurences of an event over a unit of time and assume that it has a Poisson distribution with mean $m=\lambda $. Let $T_1, T_2 , T_3, \ldots $ be the interarrival times of the occurences and they are iid with an exponential $\lambda $ distribution. Note that $ X=k $ iff $$ \sum_{j=1}^k T_j \leq 1 \quad \text{and} \quad \sum_{j=1}^{k+1} T_j >1. $$
This is precisely what I do not understand. Why does the total waiting tme until $k$ occurences have to be less than or equal to 1? Any help is greatly appreciated. Thank you.