To be more precise,
Let $f:\mathbb{R}^n \to \mathbb{R}^m \in C^k$
Let $g:\mathbb{R}^m \to \mathbb{R} \in C^k$
Let $ P $ be the Taylor Polynomial of $f$ of order k near $a \in \mathbb{R}^n$
Where $P =(P_1,...,P_m) $ and each $P_i$ is the Taylor Polynomial of the ith component of $f$
Let $ Q $ be the Taylor Polynomial of $g$ of order k near $f(a) \in \mathbb{R}^m$
Prove: The Taylor polynomial of $g \circ f$ is $ Q \circ P$ near $a$
I would rather not show this by direct computation of the coefficients, but prove its correctness by showing that $g \circ f(a+h)-Q \circ P(h)= o((\vert \vert h \vert \vert)^k) $
Progress thus far:
$g \circ f(a+h)-Q \circ P(h)= g(f(a+h)) - Q(P(h)) = g(f(a+h)) - Q(f(a+h) + o(h^k)$
I cant find a way to continue, any ideas?