Carathéodory's Theorem

Write $x = \sum_{i=1}^{k} \lambda_i x_i$ with $x_i \in S$, $\lambda_i > 0$, and $\sum_{i=1}^{k} \lambda_i = 1$, chosen so that the number of terms $k$ is minimal. We claim $k \le d + 1$.

Suppose toward a contradiction that $k \ge d + 2$. Then the $k - 1 \ge d + 1$ difference vectors $x_2 - x_1, x_3 - x_1, \ldots, x_k - x_1$ in $\mathbb{R}^d$ are linearly dependent, so there exist scalars $\mu_2, \ldots, \mu_k$, not all zero, such that $\sum_{i=2}^{k} \mu_i (x_i - x_1) = 0$. Setting $\mu_1 = -\sum_{i=2}^{k} \mu_i$ gives $\sum_{i=1}^{k} \mu_i x_i = 0$ and $\sum_{i=1}^{k} \mu_i = 0$ with at least one $\mu_i > 0$.

For any real $t$ we have $x = \sum_{i=1}^{k} (\lambda_i - t \mu_i) x_i$ with coefficients summing to $1$. Choose $t = \min_{\mu_i > 0} (\lambda_i / \mu_i) > 0$; then every new coefficient $\lambda_i - t \mu_i$ is nonnegative and at least one becomes $0$, expressing $x$ as a convex combination of at most $k - 1$ points of $S$ and contradicting the minimality of $k$.