A Convex Function of Class $C^2$ has Positive Semidefinite Hessian

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $U\subseteq\mathbb{R}^n$ be an open and convex subset of Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ and convex on $U$.

Let $0_n$ denote the real $n\times n$ matrix all of whose entries are $0$. Its entries satisfy $(0_n)_{ij}=(0_n)_{ji}$, so $0_n$ is symmetric, and the Hessian matrix $D^2f(x)$ is symmetric by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian; both therefore lie in the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, so the positive semidefinite ordering $\preceq$ applies to them.

Then for every $x\in U$,