Second-Order Taylor Expansion with Peano Remainder

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$, and let $x\in U$. Let $\mathbb{R}$ carry the operations and the order of its ordered field structure, let $|\cdot|$ be the absolute value on $\mathbb{R}$, and for $h\in\mathbb{R}^n$ write $\lVert h\rVert$ for the Euclidean distance from $h$ to the origin of $\mathbb{R}^n$, addition of points of $\mathbb{R}^n$ being the coordinatewise addition of Euclidean Space $\mathbb{R}^n$.

Then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ with $\lVert h\rVert<\delta$ satisfies $x+h\in U$ and

with the partial derivative notation of the partial derivative definition and of C^2 Real-Valued Map on an Open Subset of Euclidean Space.