After the zombies ate your brain at the end of level 1, Dr. Zomboss decides to continue sending his legions of undead until they completely destroy your house. At every 4 seconds, he drops a new zombie on some unoccupied spot of your lawn. All zombies move towards the left and each of them needs exactly 4 seconds to traverse through one of the spots on the lawn. Show that after some time T, the total distance traveled by any 1000 consecutive zombies will be almost the same - within range of just 32 spots.

HINT

You don't need to be able to point out when the time T comes, just to prove that it exists.

P.S. Unfortunately I can hardly take credit for this puzzle. I just simplified/modified a problem from some Math competition, which made the original question easily approachable. I believe that in the context of zombies, it is not that hard to figure it out (but still not that easy). Later will post reference to the original problem.