The Ants

Here's a problem Brian Kell told me, given to him by Dr. Radcliff and it's a two-parter. This is the first part of two.

A set of any number of ants are placed on the number line between 0 and 1. Each ant is given a direction, and travels at unit speed (one unit per second). Each ant is the size of a point (so has no size, just a location). When two ants run into each other, they instantly negate their velocity. Prove that no matter what arrangement of positions and directions the ants have originally they will all be outside of the interval by the end of a second.

I don't want to spoil it for you quite yet.  The solution will show up in about a week.