Homework 1

Show that the r-dimensional probability simplex

$\Delta_r = \left\{(p_0,p_1,\ldots,p_r)\in \mathbb{R}^{r+1} : \sum_{i=0}^rp_i = 1, p_i\geq 0 \mbox{ for all } i = 0,\ldots, r\right\}$

is a semialgebraic set for all $r\geq 1$.

Submit your solution by either uploading a pdf to canvas before the start of the next lecture or by giving it to me at the start of the next lecture. Be sure to write a clear, concise and well-motivated proof, as the presentation of your solution will also be considered.