In the Proposition 1.2.8 in the book "Stochastic Analysis on Manifold" by Elton Hsu, he claimed that:
Given a smooth closed embedded manifold $M$ in $\mathbb{R}^N$. Let $d_{\mathbb{R}^N}( \cdot,\cdot)$ denote the Euclidean distance between any two points in $\mathbb{R}^N$, then within a small neighborhood of $M$, the function $f(x) := \inf_{y \in M} d^2_{\mathbb{R}^N} (x,y)$ is smooth
Is this claim true? Can you help me by giving a reference or a proof on this?