Closest pair problem

1. Assume the points lie on the real number line.
2. Sort the points in ascending order.
3. Compute the difference between pairs of adjacent numbers in the sorted list.
4. Find the smallest such difference.

First, take some time to convince yourself that the above algorithm does work. Then, write an explanation that you think would be understood by a typical high school student who has never taken any computer science courses.

Next, provide the runtime of the algorithm, justifying your answer.

Finally, either translate this approach to the two-dimensional closest-pair problem or explain why the approach does not work for the two-dimensional closest-pair problem.