real analysis - Applications of Lipschitz Estimate for continuously differentiable functions?

Lipschitz Estimate for $C^1$ Functions*: Let $U \subseteq \mathbb{R}^n$ be open and $F: U \to \mathbb{R}^m$ continuously differentiable ($C^1$). Then $F$ is Lipschitz continuous on every compact convex subset of $K \subseteq U$, with Lipschitz constant $\sup\limits_{x \in K} || DF(x) ||$.

Corollary: Continuously differentiable functions $U \to \mathbb{R}^m$ are locally Lipschitz continuous.

This Lemma and its corollary are used in the proofs of the Inverse Function Theorem as well as the Fundamental Theorem for ODE's.

Question:
Are there any other (notable) examples of where this Lemma and/or its corollary are applied?

Is it only useful for constructing theoretical bounds for theoretical existence results like the Inverse Function Theorem or the Fundamental Theorem for ODE's? Or can it also give useful numerical bounds (i.e. for applications requiring explicit constructions)?

Context: I am trying to gauge how important this lemma is "by itself", e.g. outside of the context of the above two mentioned theorems. First, purely out of curiosity, and second, because I am writing up a proof of the Inverse Function Theorem, and am trying to decide whether this result should be included as "part of the proof" of the Inverse Function Theorem, or treated as an independently important and useful result, e.g. like the Banach Fixed Point Theorem.

*Proposition C.29 and Corollary C.30 in Lee's Introduction to Smooth Manifolds, Theorem 9.19 in Rudin's Principles of Mathematical Analysis.