Products and Quotients of $C^k$ Real-Valued Maps on Euclidean Open Sets Are $C^k$

Let $k,n$ be natural numbers with $k\ge 1$, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $f,g:U\to\mathbb{R}$ be maps of class $C^k$ on $U$.

(i) The pointwise product $(fg):U\to\mathbb{R}$, defined by $(fg)(x)=f(x)\,g(x)$ for $x\in U$, is of class $C^k$ on $U$.

(ii) If $g(x)\ne 0$ for every $x\in U$, then the pointwise quotient $f/g:U\to\mathbb{R}$, defined by $(f/g)(x)=f(x)/g(x)$ for $x\in U$, is of class $C^k$ on $U$.