Let be given by
Here’s the problem:
For all , does there exist some (dependent on ) for which ? (Of course, denotes iteration of , times.)
If this is true, then notice the orbit of will always converge to the sequence . Now pick a small number and try out the iterations by hand. If you’re really curious about trying some larger examples, you can open up Python and try the code at the end of this post to avoid the tedious calculations. I’ve been thinking about this problem for the past week or so and thought you’d enjoy to think about it as well.
Stay tuned for some posts on the Discrete Mathematicians Intermediate Value Theorem!