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 $\forall x \in M$ ?

Functions satisfying the first condition can be constructed easily by partition of unity and the standard technique of mollifiers. However, I can't see how to control the gradient of the approximate function.

I need this result when I'm trying to follow a proof. The existence of such a function is taken for granted in that proof, and is called the "regularization" of the distance function. This question also seems to be interesting by itself.

I'm not sure if this question is too easy for math overflow. I put it on stack exchange last week but nobody answered it. Could you please help me? Thanks a lot.