Metric limits of manifolds with positive scalar curvature

abstract: We show that any Riemannian metric conformal to the round metric on $\mathbb{S}^n$, for $n\geq 4$, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on $\mathbb{S}^n$ in the sense of uniform convergence of Riemannian distance. In particular, non-negativity of scalar curvature is not preserved under such limits.