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## Homework Statement

Let [itex]M[/itex] and [itex]N[/itex] be two metric spaces. Let [itex]f:M \to N[/itex]. Prove that a function that is

*locally Lipschitz*on a compact subset [itex]W[/itex] of a metric space [itex]M[/itex] is

*Lipschitz*on W.

A similar question was asked here

https://www.physicsforums.com/showthread.php?t=325759&highlight=locally+lipschitz+compact,

but it didn't really address my question.

## Homework Equations

Definition: a function [itex]f:M \to N[/itex] is said to be

*Lipschitz*on a set [itex]S[/itex] if there exists a positive constant [itex]L \in \Re^+ [/itex] such that [itex]\forall x,y \in S[/itex], [itex]d_N(f(x),f(y)) \leq L*d_M(x,y)[/itex]

Definition: a function [itex]f:M \to N[/itex] is said to be

*locally Lipschitz*on a set [itex]S[/itex] if for every point [itex]x_i \in S [/itex], there exists an open ball [itex]B_{r_i}(x_i)[/itex] of radius [itex]r_i[/itex] centered at [itex]x_i[/itex] such that [itex]f[/itex] is Lipschitz on [itex]B_{r_i}(x_i)[/itex] with Lipschitz constant [itex]L_i[/itex].

## The Attempt at a Solution

Because we are given that [itex]f[/itex] is locally Lipschitz, we know that [itex]\forall x_i \in W: \exists r_i \in \Re^+: f [/itex] is Lipschitz on [itex]B_{r_i}(x_i)[/itex]. The set [itex]\{B_{r_i}(x_i) | x_i \in W \}[/itex] is an open cover of [itex]W[/itex]. Since [itex]W[/itex] is compact, we can extract a finite subcover so that (after a possible re-ordering of indices of balls) [tex]W \subseteq \bigcup_{i=1}^{N}B_{r_i}(x_i) [/tex].

Now, for any two [itex]x,y \in W[/itex] there are two possibilities:

i) [itex] \exists i \in \{1,...,N\}: x,y \in B_{r_i}(x_i) [/itex]. In this case, we have [itex]d_N(f(x),f(y)) \leq L_i*d_M(x,y)[/itex]. If this were true for all pairs of points, I could simply choose a Lipschitz constant for [itex]W[/itex] to be [itex]\max_i(L_i)[/itex].

ii) [itex]\forall i \in \{1,...,N\}: x, y[/itex] are not in the same ball [itex]B_{r_i}(x_i) [/itex]. In this case, I'm really stuck regarding what to do. I have tried using the triangle inequality with little success.

This actually isn't a homework assignment. The ODE textbook I'm using stated this fact, but relegated its proof to the exercises. Any help would be appreciated