We extend the Poincare-Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space D([0, 1]). This approach is used to simplify the proof of the self-normalized Donsker theorem in Csorgo et al. (2003). Some notes on spheres with respect to lp-norms are given.