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

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?

• This property is called "lattice completeness" or "Dedekind completeness". Maybe you find a reference if you search for these terms, but it's possibly one of these results for which no one bothered to write down a proof, because they are just so simple. – MaoWao Jan 31 at 10:09

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