Second-Order Taylor Expansion with Peano Remainder

Let $n\ge1$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers with the operations, order and absolute value $|\cdot|$ of their ordered field structure, and let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ (via clause 3 there). Write $\partial_i f$ for the partial derivative of $f$ with respect to the $i$-th coordinate, and $\partial_j\partial_i f$ for the iterated partial derivative of clause 4 of C^k Maps on a Euclidean Open Set, defined on all of $U$ by clause 2 there.

Let $x\in U$. Addition of points of $\mathbb{R}^n$ is the coordinatewise addition of Euclidean Space $\mathbb{R}^n$, and for $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ we write $\lVert h\rVert$ for the Euclidean distance from $h$ to the origin of $\mathbb{R}^n$.

Then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $h\in\mathbb{R}^n$ with $\lVert h\rVert<\delta$ satisfies $x+h\in U$ and