One point compactification and Lipschitz normally embedded definable subsets

A closed subset of ${\mathbb{R}}^q$, definable in some given o-minimal structure, is Lipschitz normally embedded in ${\mathbb{R}}^q$ if and only if its one-point compactification is Lipschitz normally embedded in the unit sphere ${\mathbf{S}}^q$($={\mathbb{R}}^q\cup \{\mathrm{\infty }\}$), i.e. the closure of its image by the inverse of the stereographic projection is Lipschitz normally embedded in ${\mathbf{S}}^q$. This implies that any closed connected unbounded definable subset of an Euclidean space is definably inner bi-Lipschitz homeomorphic to a Lipschitz normally embedded definable set.