The Squared Euclidean Norm is Smooth

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, and let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^{n}$, which is an open subset of itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. For $t\in\mathbb{R}$ write $t^{2}$ for $t\cdot t$.

Then the function $q:\mathbb{R}^{n}\to\mathbb{R}$ given by $q(y)=\lVert y\rVert^{2}$ is smooth on $\mathbb{R}^{n}$.