This is a proof, and we show below that they indeed can agree the gun fires before the bullet hits the target.
Work Step by Step
In the frame, the bullet takes 1 second to hit the target, but light takes a millionth of a second to get to the target, considerably faster than the bullet. The other frame would have to travel to the right. The observer thus would have to travel the shown 300m in a millionth of a second to intercept the signals, moving faster than light. Since objects cannot move faster than light, this is impossible. Thus they can agree the gun fires first.