Say, $U, V, W$ are normed vector spaces that has differentiation. We know that if $f : U \rightarrow V$, $g : V \rightarrow W$ are differentiable, then so is $g \circ f$ and $$D (g \circ f) (x) = Dg(f(x)) \circ Df(x)$$
Question is, if $f, g$ are infinitely differentiable, is $g \circ f$ infinitely differentiable as well? Specifically, how do I go with induction step for $C^n$?