Herrero conjectured in 1991 that every multi-hypercyclic (respectively,
multi-supercyclic) operator on a Hilbert space is in fact hypercyclic
(respectively, supercyclic). In this article we settle this conjecture
in the affirmative even for continuous linear operators defined on arbitrary
locally convex spaces. More precisely, we show that, if $T : E \rightarrow E$ is a
continuous linear operator on a locally convex space E such that there is
a finite collection of orbits of T satisfying that each element in E can be
arbitrarily approximated by a vector of one of these orbits, then there is a
single orbit dense in E. We also prove the corresponding result for a weaker
notion of approximation, called supercyclicity.