International Mathematical Olympiad (IMO) $2017$ , Day $1$ , Problem $3$ of $6$

Question:

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0$ , and the hunter's starting point, $B_0$ , is the same (i.e. $A_0 = B_0$ ). After $n - 1$ rounds of the game, the rabbit is at point $A_{n - 1}$ and the hunter is at point $B_{n - 1}$ . In the $n^{th}$ round of the game, three things occur in order:

(i): The rabbit moves invisibly to a point $A_n$ such that the distance between $A_n$ and $A_{n - 1}$ is $1$ .

(ii): A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1$ .

(iii): The hunter moves visibly to a point $B_n$ such that the distance between $B_{n - 1}$ and $B_n$ is exactly $1$ .

Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds she can ensure that the distance between her and the rabbit is at most $100$ ?

The person that answers this correctly and gives the official solution first - there is only $1$ - go to International Mathematical Olympiad (IMO) Hall of Fame