There are $n$ points, $A_1$, $A_2$, $\cdots$, $A_n$ on a line segment, $\overline{A_0A_{n+1}}$. The point $A_0$ is black, $A_{n+1}$ is white, and the rest points are colored randomly either black or white. Prove: among these $n+1$ line segments $A_kA_{k+1}$, where $k=0, 1, \cdots, n$, the number of those with different colored ending points is odd.