Euclidean Open Box Criterion in $\mathbb{R}^n$

Let $n\in\mathbb{N}$ and let $U\subseteq \mathbb{R}^n$. Then $U$ is open in the Euclidean sense if and only if for every point $x=(x_1,\dots,x_n)\in U$ there exists a real number $\delta>0$ such that every point $y=(y_1,\dots,y_n)\in\mathbb{R}^n$ satisfying

for every $i\in\{1,\dots,n\}$ belongs to $U$.