  We show that for every Lipschitz function $f$ defined on a separable
Riemannian manifold $M$ (possibly of infinite dimension), for every continuous
$\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists
a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that
$|f(p)-g(p)|\leq\epsilon(p)$ for every $p\in M$ and
$\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$. Consequently, every separable
Riemannian manifold is uniformly bumpable. We also present some applications of
this result, such as a general version for separable Riemannian manifolds of
Deville-Godefroy-Zizler's smooth variational principle.
