- #1

Jamin2112

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- 12

## Homework Statement

Prove the Archimedean property

## Homework Equations

Know what a least upper bound is

## The Attempt at a Solution

Assume that if

*a*and

*b*are positive real numbers,

*na≤b*for all natural numbers n. Then the set

**S**of all numbers

*na*, where

*n*is a natural number, has

*b*as its least upper bound.

Let

*n'*be a natural number such that

*b-∂*

*< ∂n' ≤ b*. Then

*b < ∂(n'+1)*. Since

*n'*is a natural number,

*n'+1*is a natural number, and so

*∂(n'+1)*is an element of

**S**. But since

*b < ∂(n'+1)*,

**S**cannot have

*b*as its least upper bound, and we have a contradiction.