Coordinate Bounds Control the Euclidean Norm

Let $n$ be a natural number, let $[n]$ be the initial segment of $\mathbb{N}$ determined by $n$, and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$. Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm, $|\cdot|$ for the absolute value on the real numbers, whose order $\le$ is that of an ordered field, and $\iota$ for the canonical map of $\mathbb{R}$.

Let $t$ be a real number with $0\le t$ and suppose that $|x_i|\le t$ for every $i\in[n]$. Then