Chapter 7.6, Problem 18MS

The cobweb tent (ExH). Take a square in which the diagonal and an inverted V are drawn. Start at any point on the diagonal. From there, everything else is determined. Go vertically up or down as needed to head toward the inverted V. When you hit the inverted V, go horizontally right or left until you hit the diagonal. From there repeat the pattern going vertically until you hit the inverted V and then horizontally until you hit the diagonal. Repeal Following this pattern creates the cobweb plot you have seen before (see Mindscape 30 on page 539-541 in the previous section for a full description of cobweb plots). Look at the following examples. These cobweb plots are the result of a repeated process, and they illustrate many of the ideas from this section. Mindscapes 6-18 all refer to this process. You can also produce these graphs using a program on the Heart of Mathematics Web site-the images are really cool!

Target practice. Suppose we give you a target interval-for example, all the points on the diagonal above $\left[0.31,0.32\right].$

Why must there be a point on the diagonal above the interval $\left[0.0,0.01\right]$ that hits the target interval after some number of steps?