Let
![u_1, u_2, \ldots, u_m](/media/m/3/2/b/32b57ecb7384abdd01798f179740b59f.png)
be
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
vectors in the plane, each of length
![\leq 1,](/media/m/c/e/6/ce6de4c98f22bd672ca0e98f6df7771f.png)
with zero sum. Show that one can arrange
![u_1, u_2, \ldots, u_m](/media/m/3/2/b/32b57ecb7384abdd01798f179740b59f.png)
as a sequence
![v_1, v_2, \ldots, v_m](/media/m/7/a/0/7a0a07cc7e43639d71ceb61c3ce0a602.png)
such that each partial sum
![v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1, v_2, \ldots, v_m](/media/m/a/a/d/aadc2390ed72ee7ceba69626391c8855.png)
has length less than or equal to
%V0
Let $u_1, u_2, \ldots, u_m$ be $m$ vectors in the plane, each of length $\leq 1,$ with zero sum. Show that one can arrange $u_1, u_2, \ldots, u_m$ as a sequence $v_1, v_2, \ldots, v_m$ such that each partial sum $v_1, v_1 + v_2, v_1 + v_2 + v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $\sqrt {5}.$