Chapter 2, Exercise 58

The green points below show a random sequence \(\{x_n\}\). When you select a value for \(N\) on the slider, the point \(u_N\) is shown in blue and \(v_N\) is shown in orange. Note that these colors have been changed from what is in the textbook. The labels account for this. Move the slider to see how the sequences that define the limit infimum and limit supremum change. What do these dots (i.e., sequences) do as you scroll through?