9.17 Theorem Suppose f maps an open set $⊂^{n}$ into $^{m}$, and $f$ is differentiable at a point $∈$. Then the partial derivatives $_{j}_{i}()$ exist, and $$\begin{align} \mathbf{f'(x)e_j}=\sum\limits_{i=1}^m(D_jf_i)\mathbf{(x)u_i} &&(1\le j\le n)\tag{27} \end{align}$$ $_{i}$′s and $_{j}$′s are standard bases of $^{n}$ and $^{m}$ respectively.
Proof Fix . Since $f$ is differentiable at $$, $$\mathbf{f(x+}t\mathbf{e_j)}-\mathbf{f(x)}=\mathbf{f'(x)(}t\mathbf{e_j)}+\mathbf{r(}t\mathbf{e_j)}\tag{28}$$ where $∣(_{j})∣/→0$ as $→0$. The linearity of $′()$ shows therefore that $$\lim_{t\to 0}\sum\limits_{i=1}^m\frac{f_i(\mathbf{x}+t\mathbf{e_j})-f_i(\mathbf{x})}{t}\mathbf{u_i}=\mathbf{f'(x)e_j.}\tag{29}$$ It follows that each quotient in this sum has a limit, as $→0$(see Theorem 4.10), so that each $(_{j}_{i})()$ exists, and then (27) follows from (29).
Here is 4.10Theorem 4.10 and 4.14 Theorem in Rudin
My question is how we exactly use theorem 4.10 to get $(29)$? I have looked other related questions but they just mentioned this allows us to interchange limit and sum, I want to know the detail.
Thanks in advance
