Proof of Second-Order Taylor Expansion with Peano Remainder

Claim numbers refer to Elementary Order Arithmetic in an Ordered Field and to Properties of the Absolute Value in an Ordered Field as indicated. Recall from Step 0 of the proof of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian the fact, used again here, that squares are strictly monotone on nonnegative reals; we re-derive it: if $0\le\alpha$, $0\le\beta$ and $\alpha<\beta$, then $0<\beta$ by claim 2, so $\beta\alpha<\beta\beta$ by claim 10, while $\alpha\alpha\le\beta\alpha$, whence $\alpha^{2}<\beta^{2}$ by claim 2; and conversely $\alpha^{2}<\beta^{2}$ forces $\alpha<\beta$, since $\beta\le\alpha$ would give $\beta^{2}\le\alpha^{2}$.

Throughout, $\lVert z\rVert$ denotes the Euclidean distance from $z\in\mathbb{R}^{n}$ to the origin, so $\lVert z\rVert$ is the nonnegative real with $\lVert z\rVert^{2}=\sum_{i=1}^{n}z_{i}^{2}$; in particular $z_{i}^{2}\le\lVert z\rVert^{2}$ for each $i$, hence $|z_{i}|\le\lVert z\rVert$ by the monotonicity just recorded and claim 4 of Properties of the Absolute Value in an Ordered Field.

Step 1 (a path interval). Since $x\in U$ and $U$ is open, there is $r$ with $0<r$ such that every point at Euclidean distance less than $r$ from $x$ lies in $U$. Suppose $h\in\mathbb{R}^{n}$ satisfies $\lVert h\rVert<r\cdot2^{-1}$. For $\tau\in\mathbb{R}$ with $|\tau|<2$, the point $x+\tau h$ is at distance $|\tau|\,\lVert h\rVert$ from $x$, since the squared distance is $\sum_i(\tau h_i)^2=\tau^{2}\lVert h\rVert^{2}=(|\tau|\lVert h\rVert)^{2}$ and both quantities are nonnegative. By claim 10 this is less than $2\cdot(r\cdot2^{-1})=r$ when $\lVert h\rVert\ne0$, and equals $0<r$ otherwise. Hence $x+\tau h\in U$ for every $\tau$ in the open interval $(-2,2)$, which contains $0$ and $1$.

Step 2 (the path functions). Write $\partial_{i}f$ for $\partial f/\partial x_{i}$ and $\partial^{2}_{ij}f$ for $\partial^{2}f/\partial x_{i}\,\partial x_{j}$, in the notation of C^2 Real-Valued Map on an Open Subset of Euclidean Space. Since $f$ is of class $C^{2}$ it is of class $C^{1}$, and each $\partial_{i}f$ is of class $C^{1}$ on $U$. So by A Real-Valued C^1 Function is Differentiable at Every Point both $f$ and each $\partial_{i}f$ are differentiable at every point of $U$.

Define on $(-2,2)$ the functions $F(\tau)=f(x+\tau h)$ and $\Phi_{i}(\tau)=\partial_{i}f(x+\tau h)$. By Chain Rule Along an Affine Path, applied to $f$ and to each $\partial_{i}f$, these are differentiable at every $\tau\in(-2,2)$, with

Step 3 (an auxiliary function). Put $Q=\sum_{i=1}^{n}\sum_{j=1}^{n}\partial^{2}_{ij}f(x)\,h_{i}h_{j}$ and define $G:(-2,2)\to\mathbb{R}$ by

The map $\tau\mapsto F(0)+\tau F'(0)+\tau^{2}Q\cdot2^{-1}$ is differentiable at each $\tau_{0}$ with derivative $F'(0)+\tau_{0}Q$: its difference quotient at $\tau_{0}$ for an increment $k\ne0$ equals $F'(0)+(2\tau_{0}+k)Q\cdot2^{-1}$, which differs from $F'(0)+\tau_{0}Q$ by $kQ\cdot2^{-1}$, and given $\varepsilon$ with $0<\varepsilon$ one may take $\delta$ to be $\varepsilon$ if $Q=0$ and otherwise the value $\varepsilon\bigl(|Q|\cdot2^{-1}\bigr)^{-1}$, positive by claims 5 and 7, so that $|kQ\cdot2^{-1}|=|k|\,|Q|\cdot2^{-1}<\varepsilon$ by claims 4 of Properties of the Absolute Value in an Ordered Field and 10. By Derivative of a Sum and of a Difference, $G$ is therefore differentiable at every $\tau_{0}\in(-2,2)$ with

Also $G(0)=0$, and $G(1)$ is exactly the quantity whose absolute value is to be estimated, since $F(1)=f(x+h)$, $F(0)=f(x)$ and $F'(0)=\sum_{i}\partial_{i}f(x)h_{i}$.

Step 4 (two applications of the mean value theorem). By Mean Value Theorem on an Open Interval applied to $G$ on $(-2,2)$ with the points $0<1$, there is $\theta\in(0,1)$ with $G(1)-G(0)=G'(\theta)$, so $G(1)=G'(\theta)$.

For each $i$, apply Mean Value Theorem on an Open Interval to $\Phi_{i}$ on $(-2,2)$ with the points $0<\theta$: there is $\theta_{i}\in(0,\theta)$ with $\Phi_{i}(\theta)-\Phi_{i}(0)=\Phi_{i}'(\theta_{i})\,\theta$. Hence, using Step 2 and $F'(\theta)-F'(0)=\sum_{i}\bigl(\Phi_{i}(\theta)-\Phi_{i}(0)\bigr)h_{i}$,

By Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian we may replace $\partial^{2}_{ij}f(x)$ by $\partial^{2}_{ji}f(x)$, so with $R_{ji}(\tau)=\partial^{2}_{ji}f(x+\tau h)-\partial^{2}_{ji}f(x)$,

Step 5 (the estimate). Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Since $n\ge1$, the element $n\cdot n$ is positive (a sum of copies of $1$, positive by claim 6 and claim 3), so $\varepsilon'=\varepsilon\,(n\cdot n)^{-1}$ is positive by claims 7 and 5. Each $\partial^{2}_{ji}f$ is continuous at $x$, by condition 2 of C^2 Real-Valued Map on an Open Subset of Euclidean Space; taking the least of the finitely many radii by repeated use of claim 9, there is $\theta_{0}$ with $0<\theta_{0}$ such that every $z\in U$ at distance less than $\theta_{0}$ from $x$ satisfies $|\partial^{2}_{ji}f(z)-\partial^{2}_{ji}f(x)|<\varepsilon'$ for all $i,j$.

Let $\delta$ be the least, by claim 9, of $r\cdot2^{-1}$ and $\theta_{0}$, so $0<\delta$. Suppose $\lVert h\rVert<\delta$. Then $x+h\in U$ by Step 1 with $\tau=1$, and Steps 1 to 4 apply. For each $i$ we have $0<\theta_{i}<\theta<1$, so the distance from $x+\theta_{i}h$ to $x$ is $\theta_{i}\lVert h\rVert\le\lVert h\rVert<\theta_{0}$ by claim 10 and claim 2; hence $|R_{ji}(\theta_{i})|<\varepsilon'$ for all $i,j$.

Also $|h_{i}|\le\lVert h\rVert$ and $|h_{j}|\le\lVert h\rVert$, so $|R_{ji}(\theta_{i})h_{j}h_{i}|=|R_{ji}(\theta_{i})|\,|h_{j}|\,|h_{i}|\le\varepsilon'\lVert h\rVert^{2}$ by claim 4 of Properties of the Absolute Value in an Ordered Field and repeated use of claim 10 together with claim 2. Since $0<\theta<1$ we have $|\theta|\le1$, so by the triangle inequality (claim 5 of Properties of the Absolute Value in an Ordered Field) applied to the $n\cdot n$ summands,

Since $G(1)$ is the expression in the statement, this is the assertion.