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?

  • $\begingroup$ 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. $\endgroup$ – 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 $||a|-|b||\leq |a-b|$.

  • $\begingroup$ thanks, but I already know how to prove it. What I need is a reference. $\endgroup$ – John D Jan 31 at 10:01
  • 1
    $\begingroup$ @JohnD: If you have a proof, you don't need a reference :) $\endgroup$ – Mundron Schmidt Jan 31 at 10:02

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