This is the second in a series of posts that started here. In the first post I explained what I’m up to. Now let me just continue with some more questions. I’m now on to the harder Section II questions. Here’s the first one I want to look at. Even though it makes the posts shortish, I think I’m going to stick to one long question per post.
9F. Prove the Axiom of Archimedes.
Let be a real number in and let be positive integers. Show that the limit
exists, and that its value depends on whether is rational or irrational.
[You may assume standard properties of the cosine function provided they are clearly stated.]