In the answer by Jackel Frieder to this question, Relations between distance function and gradient on a Riemannian manifold (II) on the distance formula for Riemannian manifolds in terms of sup, we cover the manifold by open (convex) subsets $U_\alpha$ where we have local defined smooth functions $F_\alpha$ whose variation is bound by the distance (with Lipschitz constant no greater than 1) : $|F_\alpha(x)-F_\alpha(y)|\leq d(x,y)$
Using a partition of unity $(\Psi_\alpha)$ subordinated to the cover, it is claimed that the glued function, $F=\sum \Psi_\alpha F_\alpha$ locally is also bound by the distance.
I can easily see why it's locally bound by a multiple of the distance function (it's no surprise, since it's locally Lipschitz), but I don't get the bound (roughly, something like the Lipschitz constant being $\leq 1$).
I did try with an open neighborhood where almost all the $\Psi_\alpha$ are $0$ (locally finiteness) and things like making appear the variation of an $F_\alpha$, but I don't get the right bound (namely, $1$). Thank you for any help