Proof of the Chain Rule of multivariable functions in Munkres' Analysis on Manifolds

Question

In Munkres' Analysis on Manifolds, page 57 Theorem 7.1(the Chain Rule) it states: ...For this purpose, let us introduce the function $F(\mathbf h)$ defined by setting $F(\mathbf 0)=\mathbf 0$ and \begin{equation*} F(\mathbf h)=\frac{[\Delta(\mathbf h)-Df(\mathbf a)\cdot \mathbf h]}{|\mathbf h|} for \ 0<|\mathbf h|<\delta. \end{equation*} ...Further more, one has the equation \begin{equation*} \Delta(\mathbf h)=Df(\mathbf a)\cdot \mathbf h +|\mathbf h|F(\mathbf h) \end{equation*} for $0<|\mathbf h|<\delta$, and also for $\mathbf h=0$ (trivially). The triangle inequality implies that \begin{equation*} |\Delta (\mathbf h)|\leq m|Df(\mathbf a)|\cdot|\mathbf h|+|\mathbf h|\cdot|F(\mathbf h)|. \end{equation*}

I'm confused about $|\Delta (\mathbf h)|\leq m|Df(\mathbf a)|\cdot|\mathbf h|+|\mathbf h|\cdot|F(\mathbf h)|$. Where does this "$m$" come from? I think it should be $|\Delta (\mathbf h)|\leq |Df(\mathbf a)|\cdot|\mathbf h|+|\mathbf h|\cdot|F(\mathbf h)|$.

This a question from Munkres' Analysis on Manifolds. I show you the whole context of the question below. I have trouble understanding the statement with red line.

Answer 1

The reason is the metric used here is "sup metric". As a result, we have $|\mathbf h|=\max\{|h_1|,\cdots,|h_m|\}$ and $|Df(\mathbf a)|=\max\{|D_jf_i(\mathbf a)|;i=1,2,\cdots,n \ and\ j=1,2,\cdots,m\}$. So, we have: $$\begin{align*} |\Delta(\mathbf{h})| &=\left|Df(\mathbf{a})\cdot\mathbf{h}+|\mathbf{h}|F(\mathbf{h})\right|\\ &\leq|Df(\mathbf{a})\cdot\mathbf{h}|+\left||\mathbf{h}|F(\mathbf{h})\right|\\ &\leq m|Df(\mathbf{a})|\cdot|\mathbf{h}|+|\mathbf{h}|\cdot|F(\mathbf{h})| \end{align*}$$