Reference on Lipschitz property of the infimum of a family of Lipschitz functions

Question

I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.

However, since this is a very basic result, I am interested in a reference where it is proved.

Any suggestions?

Answer 1

N. Weaver, Lipschitz Algebras, 2018 (second edition), Proposition 1.32, p. 22.

Answer 2

You may not find this in a text but it is an easy consequence of the following: $\max\{f,g\}=\frac {f+g+|f-g|} 2$, $\min\{f,g\}=\frac {f+g-|f-g|} 2$ and $||a|-|b||\leq |a-b|$.