Hi! Thanks for the question. I am happy to explain any specific part. Which part did you think was not explained? I usually don’t go a the mathematical proof level because it might be a bit too much sometimes.

For instance I stated “If after the set of operations, the robot is still at the position (0, 0), then it is bounded”. We can prove it in a couple of ways. Let’s prove that, if after the set of operations, the robot is still at the position (0, 0), after applying the set of operation if executed N times for any N ≥ 0, the robot is at the position (0, 0). By recursion. It is true by definition for N = 0. Let’s assume it is true for N-1. We know that if the robot is at 0,0 and the robot is facing North, then after the set of operations, the robot will still be at 0,0. By symmetry, if the robot faces any other direction and is at (0,0), the robot will still be at (0,0) after the set of operations. Therefore, the robot will be at (0,0) after N sets of operations.

Consequently, the robot cannot be farter than the number of operations in the set of operations from the origin, then the robot is bounded.