- #1
Jamin2112
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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.