  We study the classic problem of fairly allocating a set of indivisible goods
among a group of agents, and focus on the notion of approximate proportionality
known as PROPm. Prior work showed that there exists an allocation that
satisfies this notion of fairness for instances involving up to five agents,
but fell short of proving that this is true in general. We extend this result
to show that a PROPm allocation is guaranteed to exist for all instances,
independent of the number of agents or goods. Our proof is constructive,
providing an algorithm that computes such an allocation and, unlike prior work,
the running time of this algorithm is polynomial in both the number of agents
and the number of goods.
