The ACLU won a lawsuit against the Squareshire Police Department, alleging that his policy of shooting suspected thieves on sight was unconstitutionally excessive force.
The argument that carried the day was that the policeman doesn't need to shoot: He runs fast enough that, if he ever catches sight of a thief he can simply pursue her on foot, and he will then never need to guess which way she's fleeing. Namely: Because the streets are of limited length, a policeman can't be more than 2 blocks away from a thief he can see, so even if he loses sight of her subsequently, he will be able to reach the point where she was last seen before she has time to reach the next intersection.
Now, however, Squareshire is growing. It consists of $N^2$ city blocks arranged in a square grid made of $2(N+1)$ streets each of length $N$. It is still patrolled by policemen who can run slightly faster than twice the speed of a suspected criminal.
You're a legal staffer with the ACLU, and you've been given the task of proving that even in the enlarged city, any policeman who spots a thief will still have an unbeatable strategy for catching up with her in finite time, and therefore he doesn't need to be allowed to shoot.
How large an $N$ can you extend this argument to?
Note 1: Your argument cannot involve calling for backup. Whether Squareshire has enough policemen that backup is always available would be a question of fact that needs to be decided by a jury, and your superiors don't want to risk that.
Note 2: In contrast to the earlier question, it is not necessary to be able to catch a thief whose position is completely unknown. Only that if the thief and the policeman are ever present in the same street, then she won't afterwards be able to escape.