Why is the projection from the standard simplex to the sphere Lipschitz continuous?

Question

In the book by Bridson and Häfliger, the following question is presented:
Prove that the identity map of any abstract simplicial complex induces a bi-Lipschitz homeomorphism between any two regular $M_\kappa$-simplicial complexes associated with $K$ (where $\kappa$ is not fixed).
I am interested in the case where the map goes between a possibly infinite-dimensional euclidean simplex and its spherical variant. The induced map in question would then be the projection from the simplex to the unit sphere. It is easy to see that the Euclidean distance is smaller than the corresponding spherical distance. However, I struggle to prove the other inequality, namely, that the projection is Lipschitz continuous.