Lipschitz continuity of vector-valued function

I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by ##\pmb{f}=\pmb{f}(t,\pmb{x})##, where ##\pmb x## is a vector in ##\mathbf{R}^n## that depends on ##t##.

Lemma. Assume that ##\Omega\subseteq \mathbf{R}\times\mathbf{R}^n## is a convex and bounded set, and that the function ##\pmb{f}## is continuously differentiable in a neighborhood of ##\overline{\Omega}##. Then ##\pmb{f}## is Lipschitz continuous in ##\Omega##.

Proof. The line segment between two points ##(t,\pmb{x})## and ##(t,\pmb{y})## in ##\Omega## is contained in ##\Omega## by the assumption of convexity. Hence \begin{align} \pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})&=\int_0^1\frac{\mathrm{d}}{\mathrm{d}s}\pmb{f}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))\mathrm{d}s \tag1 \\ &=\int_0^1\sum_{i=1}^n\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))(x_i-y_i)\mathrm{d}s,\tag2\end{align} by the chain rule. Set ##K=\max_i\sup_{\Omega}\left|\frac{\partial f}{\partial x_i}\right|## (which exists by assumption). Then $$|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq\int_0^1 nK|\pmb{x}-\pmb{y}|\mathrm{d}s=nK|\pmb{x}-\pmb{y}|.\tag3$$

Here's my understanding of the notation. In ##(2)##, ##\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))## is a vector evaluated at ##(t,\pmb{y}+s(\pmb{x}-\pmb{y}))##. ##\frac{\partial}{\partial x_i}## denotes the partial derivative of ##\pmb{f}(t,x_1,\ldots,x_n)## with respect ##x_i## (a better notation in ##(2)## would probably be ##\frac{\partial}{\partial z_i}##, as ##\pmb{x}## is already being used to denote a vector). More precisely, if ##f_1,\ldots,f_n## are the components of ##\pmb{f}##, then $$\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))=\begin{pmatrix}
\frac{\partial f_1}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y})) \\ \vdots \\ \frac{\partial f_n}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))
\end{pmatrix}.\tag4$$

My question is that I do not really understand how ##(3)## is obtained. For the sake of brevity, I will write ##\frac{\partial \pmb{f}}{\partial x_i}## when I mean ##\frac{\partial \pmb{f}}{\partial x_i}(t,\pmb{y}+s(\pmb{x}-\pmb{y}))##. Suppose ##n=2##. Taking the norm (##\ell^2## norm to be exact) of ##(1)## and using ##\left\lVert\int_a^b \pmb{f}\right\rVert\le\int_a^b\left\lVert\pmb{f}\right\rVert##, we get
\begin{align}
\lVert\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})\rVert_2&=\left\lVert\int_0^1\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i)\mathrm{d}s\right\rVert_2 \tag5\\
&\leq \int_0^1\left\lVert\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i) \right\rVert_2 \mathrm{d}s \tag6
\end{align}
Next, using the triangle inequality, the norm inside the integral gets bounded by
\begin{align}
\left\lVert\sum_{i=1}^2\frac{\partial \pmb{f}}{\partial x_i}(x_i-y_i) \right\rVert_2 &\leq\left\lVert \frac{\partial \pmb{f}}{\partial x_1}(x_1-y_1)\right\rVert_2+\left\lVert \frac{\partial \pmb{f}}{\partial x_2}(x_2-y_2)\right\rVert_2. \tag7
\end{align}
The above terms on the right-hand side equal
\begin{align}
\sqrt{\left(\frac{\partial f_1}{\partial x_1}\right)^2+\left(\frac{\partial f_2}{\partial x_1}\right)^2}|x_1-y_1|+\sqrt{\left(\frac{\partial f_1}{\partial x_2}\right)^2+\left(\frac{\partial f_2}{\partial x_2}\right)^2}|x_2-y_2|.\tag8
\end{align}
Letting ##K=\max\limits_{i,j\in\{1,2\}}\sup\limits_{\Omega}\left|\frac{\partial f_i}{\partial x_j}\right|##, ##(8)## gets bounded by
\begin{align}
\sqrt{2}K|x_1-y_1|+\sqrt{2}K|x_2-y_2|=\sqrt{2}K(|x_1-y_1|+|x_2-y_2|) \tag9
\end{align}
which is clearly not equal to ##nK|\pmb{x}-\pmb{y}|##. I suspect I am doing something wrong, but I'm unable to spot it.

A basic upper bound for the value of an integral of [itex]F: [a,b] \to \mathbb{R}[/itex] is [tex]
\int_a^b F(s)\,ds \leq (b- a)\sup_{s \in [a,b]} F(s).[/tex] By extension, given [tex]I = \int_0^1 \sum_{i=1}^n\|\mathbf{F}_i(t,s)\||x_i - y_i|\,ds[/tex] we can obtain [tex]\begin{split}
I &\leq \sum_{i=1}^n \sup_s \|\mathbf{F}_i\||x_i - y_i| \\
&\leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) \sum_{i=1}^n |x_i - y_i|.\end{split}[/tex] Now [tex]
\begin{split}
\sum_{i=1}^n |x_i - y_i| &= \left|(1, \dots, 1) \cdot (|x_1-y_1|, \dots, |x_n - y_n|)\right| \\
& \leq \| (1, \dots, 1) \| \|(|x_1-y_1|, \dots, |x_n - y_n|)\| \\
& = \sqrt{n} \|\mathbf{x} - \mathbf{y}\| \\
& \leq n \|\mathbf{x} - \mathbf{y}\|.\end{split}[/tex] so that [tex]
I \leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) \sum_{i=1}^n |x_i - y_i| \leq \left(\max_i \sup_s \|\mathbf{F}_i\|\right) n \|\mathbf{x} - \mathbf{y}\|.[/tex]

Thank you for replying.

In the statement of the lemma, the authors stipulate that ##\pmb f## be ##C^1## in a neighborhood of the closure of ##\Omega## (I interpret ##\overline{\Omega}## to be the closure of ##\Omega##). Do you by any chance know why this is required? I see why we need the function to be ##C^1##, since we are integrating its derivative, but I do not see why this condition needs to be satisfied in the closure of ##\Omega##.

To guarantee boundedness of the derivatives we require that they be continuous (hence [itex]C^1[/itex]) on a compact domain. This is the generalisation of the theorem that a function continuous on a closed, bounded interval is bounded, but a function continuous on an unclosed bounded interval may not be bounded (eg. [itex]x^{-1}[/itex] on [itex](0,1][/itex]).

For the Euclidean metric on [itex]\mathbb{R}^m[/itex], a subset is compact if and only if it is closed and bounded. We are given that [itex]\Omega \subset \mathbb{R} \times \mathbb{R}^n \simeq \mathbb{R}^{n+1}[/itex] is bounded, but we are not told that it is closed; its closure, however, is certainly closed and bounded.