In Robert Wald's General Relativity, he states that
If $F: \mathbb{R}^n \to \mathbb{R}$ is $C^\infty$, then for each $a=(a^1, \dots, a^n) \in \mathbb{R}^n$ there exist $C^\infty$ functions $H_\mu$ such that for all $x \in \mathbb{R}^n$ we have $$ F(x) = F(a) + \sum_{\mu = 1}^n (x^\mu - a^\mu)H_\mu(x) $$ Furthermore, we have $$ H_\mu(a) = \frac{\partial F}{\partial x^\mu}\Bigr|_{x=a}$$
Where the superscripts $x^\mu$ indicate indices, not exponents. Problem 2 at the end of the chapter then gives a hint to prove this by induction:
(Hint: For $n=1$, use the identity $$ F(x) - F(a) = (x-a) \int_0^1 F'[t(x-a) + a]dt$$ then prove it for general $n$ by induction.)
I've seen other ways of proving the statement, but for the life of me I can't see how to prove it by induction. Any insight would be greatly appreciated!
The base case proceeds as follows. For a smooth function $F: \mathbb{R} \to \mathbb{R}$ and some $a \in \mathbb{R}$, we can construct $H(x) = \frac{F(x) - F(a)}{x-a}$ so that $F(x) - F(a) = (x-a)H(x)$ and $H(a) = \frac{d}{dx}F(x) \Bigr|_{x=a}$. Such an $H$ is smooth and well-defined on $\mathbb{R}$ because $F$ is smooth.
Now suppose the proposition is true for $n<k-1$. Given a smooth function $F: \mathbb{R}^k \to \mathbb{R}$ and some $a \in \mathbb{R}^k$, we can consider fixing the last coordinate $x^k = a^k$, letting us apply the inductive proposition
$$ F(x^1, \dots, x^{k-1}, a^k) = F(a) + \sum_{\mu=1}^{k-1} (x^\mu - a^\mu) H_\mu(x^1, \dots, x^{k-1})$$
Now if we define
$$ H_k(x) \equiv \frac{F(x) - F(x^1, \dots, x^{k-1}, a^k)}{x^k - a^k} $$
We find that $H_k$ is smooth and well-defined on $\mathbb{R}^k$ again because $F$ is smooth, and we recover the intended proposition
$$ F(x) - F(a) = \sum_{\mu=1}^{k-1} (x^\mu - a^\mu) H_\mu(x^1, \dots, x^{k-1}) + (x^k - a^k)H_k(x) = \sum_{\mu=1}^{k} (x^\mu - a^\mu) H_\mu(x)$$
and
$$ H_k(x) = \frac{\partial}{\partial x^k} F(x) \Bigr|_{x=a} $$