A Convex Function is Lipschitz on a Ball around an Interior Point

Let $n$ be a natural number with $1\le n$, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, with the absolute value written $|\cdot|$.

On Euclidean space $\mathbb{R}^{n}$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $\lVert\cdot\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean metric, which satisfy $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; closed balls $\bar{B}_{d_{E}}(x,s)$ are those of Closed Ball in a Metric Space.

Regard $\mathbb{R}^{n}$ as a topological space with the collection of those subsets that are open in the metric space $(\mathbb{R}^{n},d_{E})$, a topology by Metric Open Sets Form a Topology.

Let $C\subseteq\mathbb{R}^{n}$ be convex, let $u:C\to\mathbb{R}$ be convex on $C$, and let $x_{0}$ be an interior point of $C$ in $\mathbb{R}^{n}$.

Then there are $\rho,L\in\mathbb{R}$ with $0<\rho$ and $0\le L$ such that $\bar{B}_{d_{E}}(x_{0},\rho)\subseteq C$ and

$$\bigl|u(y)-u(x)\bigr|\le L\,\lVert y-x\rVert\qquad\text{for all }x,y\in\bar{B}_{d_{E}}(x_{0},\rho).$$