Local uniformization and arc spaces

Let X be a variety over a perfect field k and let X∞ be its space of arcs. Given a closed subset Z of X, let X∞Z denote the subscheme of X∞ consisting of all arcs centered at some point of Z. We prove that Local Uniformization implies that X∞Z has a finite number of irreducible components for each closed subset Z of X. In particular, Local Uniformization implies that X∞SingX has a finite number of irreducible components.