Good evening, I'm trying to do this exercise in preparation for the mid-term exam in analysis:
Let $X$ be open in $\mathbb R^n$, $f:X \to F$, and $m \in \mathbb N$. Assume that $\partial_{j_1} \partial_{j_2} \cdots \partial_{j_{m+1}} f$ is continuous at $x$ for all $j_1, j_2, \ldots, j_{m+1} \in \{1,\ldots,n\}$ and $\partial^m f(x)$ is also continuous at $x$. For $j \in \{1,\ldots,n\}$, give the formula of $\partial_j (\partial^m f)(x)$ in terms of $\partial_{j_1} \partial_{j_2} \cdots \partial_{j_{m+1}} f(x)$.
Honestly, my proof is complicated with many formulas but I'm unable to reduce it anymore. The algebra of multilinear maps is a burden to me. This exercise is very important to me because it tests if I correctly understand the concepts of derivative and partial derivatives and their relationship.
I'm unable to determine if changing the order between $\lim_{t \to 0}$ and $\sup$ is right or wrong.
$$\begin{aligned}& \lim_{t \to 0} \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \|\\ \le & \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \lim_{t \to 0} \Bigg[ \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} \Bigg] - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \| \\ \end{aligned}$$
As such, I very hope that somebody verifies my proof and give suggestions on it. Thank you so much for your help!
$\textbf{My attempt:}$
Let $\{e_1, \ldots,e_n\}$ be the standard basis of $\mathbb R^n$. We have $$\partial_{j_1} \partial_{j_2} \cdots \partial_{j_m} f (x) = \partial^m f (x) [e_{j_1}, e_{j_2}, \ldots, e_{j_m}]$$ and so $$\begin{aligned} \partial^m f (x) [h^1, \ldots, h^m] &= \partial^m f (x) \left [\sum_{i=1}^n h_{i}^1 e_i, \ldots, \sum_{i=1}^n h_i^m e_i \right] \\ &= \sum_{j_1, \ldots, j_m =1}^n \partial^m f (x) \left [h^1_{j_1} e_{j_1}, \ldots, h^m_{j_m} e_{j_m} \right] \\ &= \sum_{j_1, \ldots, j_m =1}^n \partial^m f (x) \left [ e_{j_1}, \ldots, e_{j_m} \right] (h^1_{j_1} \cdots h^m_{j_m}) \\ &= \sum_{j_1, \ldots, j_m =1}^n \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \end{aligned}$$
for all $h^i = (h_1^i, \ldots, h_n^i) \in \mathbb R^n$ with $1 \le i \le m$. For each $x \in X$, we define a map $A_x$ by $$A_x: \mathbb R^m \to F, \quad A_x [h^1, \ldots,h^m] \mapsto \sum_{j_1, \ldots, j_m =1}^n \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) $$
We have $$\begin{aligned} & \left \| \frac{\partial^m f(x +te_j) - \partial^m f(x)}{t} - A_x\right \| \\ =& \sup_{|h^1|,\ldots,|h^m| \le 1} \left \| \frac{\partial^m f(x +te_j) - \partial^m f(x)}{t} [h^1, \ldots,h^m] - A_x [h^1, \ldots,h^m] \right \| \\ =& \sup_{|h^1|,\ldots,|h^m| \le 1} \left \| \frac{\partial^m f(x +te_j)[h^1, \ldots,h^m] - \partial^m f(x) [h^1, \ldots,h^m]}{t} - A_x [h^1, \ldots,h^m] \right \| \\ =& \sup_{|h^1|,\ldots,|h^m| \le 1} \Bigg \| \frac{ \sum_{j_1, \ldots, j_m =1}^n \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \quad \quad \quad\quad \frac{- \sum_{j_1, \ldots, j_m =1}^n \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \quad \quad \quad\quad - \sum_{j_1, \ldots, j_m =1}^n \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \| \\ =& \sup_{|h^1|,\ldots,|h^m| \le 1} \Bigg \| \sum_{j_1, \ldots, j_m =1}^n \Bigg [ \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg ]\Bigg \| \\ \le& \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \|\\ \end{aligned}$$
Hence
$$\begin{aligned} & \lim_{t \to 0} \left \| \frac{\partial^m f(x +te_j) - \partial^m f(x)}{t} - A_x\right \| \\ \le & \lim_{t \to 0} \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \|\\ \le & \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \lim_{t \to 0} \Bigg[ \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) (h^1_{j_1} \cdots h^m_{j_m})}{t} \\ & \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m})}{t} \Bigg] - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \| \\ \le & \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| (h^1_{j_1} \cdots h^m_{j_m}) \cdot \lim_{t \to 0} \Bigg[ \frac{ \partial_{j_1} \cdots \partial_{j_m} f (x + te_j) }{t} \\ & \quad \quad \frac{- \quad \partial_{j_1} \cdots \partial_{j_m} f (x)}{t} \Bigg] - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \|\\ = & \sup_{|h^1|,\ldots,|h^m| \le 1} \sum_{j_1, \ldots, j_m =1}^n \Bigg \| \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (x) (h^1_{j_1} \cdots h^m_{j_m}) \Bigg \|\\ =& \quad 0 \end{aligned}$$
As such, $\partial_j (\partial^m f)(x) = A_x$.