Narrow convergence and convergence of points

Question

I'm reading a paper about the curvature-dimension condition in Lorentzian geodesic spaces, and there's this line: "Let $\mathfrak{m}_j\rightharpoonup\mathfrak{m}_\infty$ be probability measures converging narrowly (the ambient space is a proper metric space). Then from that we deduce the existence of a sequence $x_j\in\mathrm{supp}\,\mathfrak{m}_j$ with $x_j\rightarrow x_\infty\in\mathrm{supp}\,\mathfrak{m}_\infty$." Does this come from the convergence of the measures (that implies some sort of convergence of the supports?)?