A Local Minimum of a Convex Function is Global

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the ordered field of real numbers. Equip Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the Euclidean distance $d_E$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$; local minima refer to $d_E$.

Let $C\subseteq\mathbb{R}^n$ be convex, let $f:C\to\mathbb{R}$ be convex on $C$, and let $x_0\in C$.

If $f$ has a local minimum at $x_0$ relative to $C$, then