  Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a
random process that takes values in the vertex set of a graph G, which is more
likely to visit vertices it has visited before. Pemantle and Volkov considered
the case when the underlying graph is the one-dimensional integer lattice Z.
They proved that the range is almost surely finite and that with positive
probability the range contains exactly five points. They conjectured that this
second event holds with probability 1. The proof of this conjecture is the main
purpose of this paper.
