Carathéodory's Theorem in $\mathbb{R}^n$

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, with the convex hull $\operatorname{conv}$ and the standard simplices $\Delta_{N}$, let $S\subseteq\mathbb{R}^{n}$.

(i) At most $n+1$ points. Every $z\in\operatorname{conv}S$ is a convex combination of $N$ points of $S$ for some $N\in\mathbb{N}$ with $N\le n+1$: there are $x_{1},\dots,x_{N}\in S$ and $t\in\Delta_{N}$ with

$$z=\sum_{k=1}^{N}t_{k}x_{k}.$$