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:

  1. \frac{1}{n} \rightarrow 0 as n \rightarrow \infty
  2. \not\exists \epsilon. \forall n. 0 < \epsilon < \frac{1}{n}
  3. \not\exists M.\forall n.n < M
by letting
M = \frac{1}{\epsilon}

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

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