Let $X$ be the vector space of Lipschitz-Continuous functions on $[0,1]$
Show that $\vert \vert x \vert \vert_{\infty} \leq \vert \vert x \vert \vert_{\operatorname{Lip}}$
where $\vert \vert x \vert \vert_{\operatorname{Lip}}=\vert x(0)\vert+\sup\limits_{s\neq t}\vert \frac{x(t)-x(s)}{t-s}\vert$
My idea: Let $u \in [0,1]$ arbitrary and since $x$ is a continuous function
$x(u)=x(0)+\frac{x(u)-x(0)}{u-0}u$ (**) and therefore
$\vert x(u)\vert\leq\vert x(0)\vert+\vert\frac{x(u)-x(0)}{u-0}\vert u\leq \vert x(0)\vert+\vert\frac{x(u)-x(0)}{u-0}\vert\leq \vert x(0)\vert +\sup\limits_{s\neq t}\vert \frac{x(t)-x(s)}{t-s}\vert=\vert \vert x \vert \vert_{\operatorname{Lip}}$
I think I have the right idea of the proof but I am not sure on (**) and would like to see if my proof is correct. I have not used Lipschitz-Continuity which makes me believe I am wrong. Thank you for your help.