Product of functions of class $C^r$ 
Let $f,g:\mathbb{R}^n\rightarrow\mathbb{R}$ be of class $C^r$. Show that the function $h(\textbf{x})=f(\textbf{x})g(\textbf{x})$ is of class $C^r$.

I'm posting this as a reference for myself, so I'm answering my own question.
 A: So I'm looking at $r=1$ first. That $f$ is of class $C^1$ means that for each $i=1,2,\ldots,n$, $$\lim_{t\rightarrow 0}\dfrac{f(\textbf{a}+t\textbf{e}_i)-f(\textbf{a})}{t}$$ exists and is continuous. The same goes for $g$. Then we must prove  the same for $h=fg$. The limit in question may be rewritten as $$\lim_{t\rightarrow 0}\dfrac{f(\textbf{a}+t\textbf{e}_i)(g(\textbf{a}+t\textbf{e}_i)-g(\textbf{a}))+g(\textbf{a})(f(\textbf{a}+t\textbf{e}_i)-f(\textbf{a}))}{t}$$ which is equal to $$f(\textbf{a})\lim_{t\rightarrow 0}\dfrac{g(\textbf{a}+t\textbf{e}_i)-g(\textbf{a})}{t}+\lim_{t\rightarrow 0}\dfrac{f(\textbf{a}+t\textbf{e}_i)-f(\textbf{a})}{t}g(\textbf{a}) = f(\textbf{a})D_ig(\textbf{a})+D_if(\textbf{a})g(\textbf{a})$$ so it exists and is continuous.
Now perform induction on $r$. Assume the statement holds for values less than $r$, and we'll prove it for $r$. We must show that $D_ih$ is of class $C^{r-1}$ for all $i$. But we know $D_ih(\textbf{a})=f(\textbf{a})D_ig(\textbf{a})+D_if(\textbf{a})g(\textbf{a})$. These terms are of class $C^{r-1}$, so by the induction hypothesis, their product is $C^{r-1}$, and their sum is also $C^{r-1}$ (not hard to prove), so the total is $C^{r-1}$, as desired.
