A Continuously Differentiable Map Whose Jacobian Matrix is Close to the Identity Maps a Cube into a Slightly Larger Cube

In the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d$ be a natural number with $1\le d$, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background (i) where a real number is required; $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space Is Open in Itself, and $\delta_{ij}$ denotes the entry in row $i$ and column $j$ of the identity matrix $I_{d}$. For $c\in\mathbb{R}^{d}$ and a nonnegative real number $r$ let

$$Q(c,r)=\{x\in\mathbb{R}^{d}:\ |x_{i}-c_{i}|\le r\ \text{for every }i\in[d]\}$$

be the closed cube with centre $c$ and half-side $r$. Let $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a map whose components $G_{1},\dots,G_{d}$ are of class $C^{1}$ on $\mathbb{R}^{d}$.

(Cubes)(i) Let $c\in\mathbb{R}^{d}$, let $r$ and $\eta$ be nonnegative real numbers, and suppose that

$$|\partial_{j}G_{i}(x)-\delta_{ij}|\le\frac{\eta}{d}\qquad\text{for every }x\in Q(c,r)\text{ and all }i,j\in[d].$$

Then $G(x)\in Q\bigl(G(c),(1+\eta)\,r\bigr)$ for every $x\in Q(c,r)$.