Question about Lipschitz continuity

[quasar987]

1. The problem statement, all variables and given/known data
This should be easy but I'm stomped.

Let K be a compact set in a normed linear space X and let f:X-->X be locally Lipschitz continuous on X. Show that there is an open set U containing K on which f is Lipschitz continuous.


2. Relevant equations
locally Lipschitz means that for every x in X, there is a nbdh around x on which f is Lipschitz.


3. The attempt at a solution
The obvious thing to do it seems it take an open cover of K by sets on which f is Lipschitz continuous and extract a finite subcover. But then what?!?

[Dick]

Wouldn't it be nice if you could make sure that the final open set containing K and covered by the subcover, were convex? Replace K by it's convex hull, it's still compact, right? Then take the finite subcover. Then pick a open set just a 'little' bigger than K, but still convex and contained in the subcover. That works, doesn't it?

[quasar987]

Of course, yes! Thanks Dick.