This applet is designed to help you understand why some iterative sequences for solving equations do converge to a solution (if the circumstances are right), while others do not.
You can move the graph by clicking on the dot at its vertex, and dragging it with your mouse. You can move the starting position of the iteration by clicking and dragging the dot on the x-axis. By dragging the slider (at the bottom of the page) you can change the shape of the curve further. Try it and see.

By playing around you should see that sequences sometimes converge to a solution by circling it, getting closer all the time (cobwebs) or by climbing up or down stepped sequences (staircases), or a combination of both. What are the conditions necessary for either of these to happen? Look carefully and try to work it out.
Convergence of sequences is a large and tricky area of mathematics, but this applet should help you get some insight into it. Enjoy!