Is the Cartesian product of $C^\infty$ functions a $C^\infty$ function?

Question

Define the Cartesian product of two functions $f:\mathbb{R}^a\to\mathbb{R}^b$ and $g:\mathbb{R}^c\to\mathbb{R}^d$ as $$(f\times g)(x,y)=(f(x),g(y)).$$ If the function $f$ and $g$ are $C^\infty$, is the function $f\times g$ also $C^\infty$?

Answer 1

The following argument shows that $f\times g$ is $C^1$ if $f$ and $g$ are both $C^1$, and then induction and be applied to prove that it is $C^\infty$ if they are $C^\infty$.