Exercise 6, § 2

Let U be an open set of a space $\mathbf{R}^n$. A path $c:\mathbf{I}\rightarrow U$ is said to be piecewise affine if there exists a finite sequence $(t_0, . . . , t_n)$ such that $0 =t_0< t_1<\cdots < t_n= 1$ and such that the restriction of $c$ to each interval $[t_{i-1}, t_i]$ is affine. Exercise prove that every path in U is strictly homotopic to a piecewise affine path.