Firstly, the Assignment:
Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function:
$$f: K_1(0) \rightarrow \mathbb{R}^3,\begin{pmatrix}x\\y\\z\\\end{pmatrix} \rightarrow \begin{pmatrix}\frac{1}{8}(x+e^y+e^z)\\\frac{1}{8}(x+y+e^z)\\\frac{1}{8}(x+y+z)\\\end{pmatrix}$$ Show that $f$ is Lipschitz and determine a Lipschitz constant.
Hint: Use common tools from differentiation to get the inequalities you need.
What I know is that I need to show that for arbitary $x, x' \in K_1(0)$: $\|f(x) - f(x')\| ≤ \lambda \cdot \|x-x'\|$. I'm pretty sure (with the hint) that I will use the MVT eventually, but I don't know how to simplify the $\|$. For $\|x-x'\|$ I thought about assume that the difference between the first two entries are the maximum, since it'd make no difference later, but I don't know how to determine which entry of the functions values' differences is the maximum.
Any help would be appreciated.