Analysis | Part IA, 2003

Prove the Axiom of Archimedes.

Let xx be a real number in [0,1][0,1], and let m,nm, n be positive integers. Show that the limit

limm[limncos2n(m!πx)]\lim _{m \rightarrow \infty}\left[\lim _{n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right]

exists, and that its value depends on whether xx is rational or irrational.

[You may assume standard properties of the cosine function provided they are clearly stated.]

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