The Gradient of a $C^2$ Function on Euclidean Space is One-Sided Lipschitz on Bounded Sets

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number and let $V:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{n}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $B\subseteq\mathbb{R}^{n}$ be bounded.

Then there is a nonnegative $c\in\mathbb{R}$ such that

where $\lVert x-y\rVert^{2}=\lVert x-y\rVert\lVert x-y\rVert$.