Talk:Axiom of Archimedes

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The statement Showing that the natural numbers are unbounded in the reals is equivalent to showing:


 * $$\frac{1}{n} \rightarrow 0$$ as $$n \rightarrow \infty $$

is accurate, although I cannot produce a citation, as it's been 30 years since I've cracked a real analysis textbook.

Proof:

The following statements are equivalent, in order:
 * $$\frac{1}{n} \rightarrow 0$$ as $$n \rightarrow \infty $$
 * $$\not\exists \epsilon. \forall n. 0 < \epsilon < \frac{1}{n}$$
 * $$\not\exists M.\forall n.n < M$$
 * by letting
 * $$M = \frac{1}{\epsilon}$$

Arthur Rubin | (talk) 15:07, 16 February 2006 (UTC)