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Problem 83: (Equivalent Forms Of The Archimedean Property)

Prove that following statements are equivalent.

  1. The set of natural numbers is unbounded above in $\mathbb{R}$.
  2. For each $z\in\mathbb{R}$, there exists $n\in\mathbb{N}$ such that $n>z$.
  3. For each real number $x>0$ and for each $y\in \mathbb{R}$, there exists $n\in\mathbb{N}$ such that $nx>y$.
  4. For each real number $x>0$, there exists $n\in \mathbb{N}$ such that $0< 1/n<x$.
Remember you are not trying to prove each statement is true, rather you are trying to prove that the statements are equivalent. So prove that if statement 1 is true, then statement 2 must be true. Then prove that if statement 2 is true, then statement 3 must be true. Then prove that if statement 3 is true, then statement 4 must be true. Finally prove that statement 4 implies statement 1.

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