This is a game played by Jr. Proofini and a fox, and it takes place on the set Z x Z, which we call \the integer lattice". Jr. Proofini is trying to catch the fox. - Before the game begins, the fox chooses two secret integer numbers a and b. - The fox is not allowed to change these numbers during the game. - Jr. Proofini does not know what a and b are. - During the night, the fox moves from its current position (x; y) to the position (x + a, y + b), and hides there until the next night. - During the day, Jr. Proofini chooses any one integer lattice point (x, y) she wants, and searches for the fox. - If Jr. Prooni searches on the exact grid point that the fox is on, then she has caught the fox! - The game begins on the first night with the fox moving to (a, b). Prove that no matter what integers a, b the fox picks to start the game, Jr. Proofini can always catch the fox after finitely many days. (Hint: As a warm-up, first solve the question assuming that a = 0 and b is either 1 or -1.)

(Hint: Cardinality, Bijection,..)