I am reading a proof of a basic result in differential geometry, which I am having trouble following.
The statement is the following: Suppose $X$ is a vector field on $M$, a $C^{\infty}$ manifold, and $(U; x_1, ..., x_n)$ be a coordinate chart. Then $X|_U$ can be uniquely expressed as $$ X|_{U} = \sum_{j=1}^n a_j \frac{\partial}{\partial x_j} $$ where $a_j \in C^{\infty}(M)$.
Proof presented goes as follows: Define $a_j = X x_j.$ Let $ f \in C^{\infty}(U)$ and $p \in U$. Then we may write
$$
f = f(p) + \sum_i \frac{\partial f}{\partial x_i} (p) (x_i - x_i(p)) + h,
$$
where $h$ is a linear combination of products $h_1h_2$ with $h_1(p) = h_2(p) = 0$.
Then since $X$ is a derivation
$$
X(h_1 h_2)(p) = h_1(p) X h_2(p) + h_2(p) Xh_1(p) =0.
$$
From which point the statement is easy enough to deduce.
My question is: We only know $f$ is in $C^{\infty}(U)$. How does one deduce about $h$ only from this information? I am having difficulty following this because $f$ may not have a converging Taylor series in which case I don't see how one can deduce that $h$ is a linear combination of products $h_1h_2$ with $h_1(p) = h_2(p) = 0$ (I can see this if we assume $f$ has a Taylor series expansion everywhere... but not sure how to see this here).
Thank you.
The standard proof is to write $$f(x)=f(p) + \sum g_i(x)(x_i-x_i(p))$$ where $g_i$ is smooth and $g_i(p) = \partial f/\partial x_i(p)$. Working in $\Bbb R^n$ with $p=0$, just write $$f(x) = f(0) + \int_0^1 \frac d{dt} f(tx)\,dt$$ and use the chain rule.