Regularized distance function on Riemannian manifolds Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let $\epsilon>0$ be an arbitrary positive number. Is there a smooth function $\tilde{d}_p(x)$ on $M$, such that 
$$ | d_p(x)-\tilde{d}_p(x) | < \epsilon$$
$$ |\textrm{grad}(\tilde{d}_p)(x)|<2$$
for $\forall x \in M$ ?
I need this result when I'm trying to follow a proof. The existence of such a function seems to be taken for granted in that proof, and is called the "regularization" of the distance function. 
Functions satisfying the first condition can be constructed easily by using the partition of unity and the standard technique of mollifiers. However, I can't see how to control the gradient of the approximate function. Could you please help me? Thanks a lot.

Note that $d_p$ is a $1$-lipschitz function. Then theorem 1 in this article 
http://www.mat.ucm.es/~dazagrar/articulos/AFLRjmaa.pdf
tells you that you may approximate it by a smooth function satisfying the properties you give (note that the gradient estimate is satisfied whenever the approximating function has lipschitz constant less than $2$).
