Chapter 2, Exercise 4

The tool for this exercise shows the elements of the sequence \(x_n = 1 - \frac{1}{n}\), the limit value \(L = 1\), and the \(\epsilon\)-band \((L−\epsilon,L+\epsilon)\) about \(L = 1\). The slider allows you to change \(\epsilon\). Doing this causes the image to zoom in vertically (by changing the limits on the vertical axis) and zoom out horizontally (by changing the limits on the horizontal axis). Use the tool to estimate the value of \(N\) needed to ensure that all further elements of the sequence are in the \(\epsilon-\) band. Do this for 5 different values of \(\epsilon\). Do you see a relationship between your chosen value of \(\epsilon\) and \(N\)?.