Using the Archimedean Property, and the Completeness Axiom, provethat given b >

Question

Using the Archimedean Property, and the Completeness Axiom, provethat given   b > 0, a real number, and any real > 0, we can find an integer N suchthat b/N < .
Note: the idea is that we can find this N even when b isreally large and is small....


Archimedean Property of the RealNumbers: If  a and b arepositive real numbers, then there exists a poitiveinterger  n suchthat na> b.

Completeness Axiom: Each nonempty set of real numbersthat is bounded above has a supremum. a)If y is a rational number such that y2 >2, then y is an upper bound of S. b)Every rational number that is an upper bound of S is greater than1. c)The number q is rational. Using the Archimedean Property, and the Completeness Axiom, provethat given   b > 0, a real number, and any real > 0, we can find an integer N suchthat b/N < .
Note: the idea is that we can find this N even when b isreally large and is small....


Archimedean Property of the RealNumbers: If  a and b arepositive real numbers, then there exists a poitiveinterger  n suchthat na> b.

Completeness Axiom: Each nonempty set of real numbersthat is bounded above has a supremum. a)If y is a rational number such that y2 >2, then y is an upper bound of S. b)Every rational number that is an upper bound of S is greater than1. c)The number q is rational.

Explanation / Answer

Choose N>b/max{,1}. Thenb/N<b/(b/)=.

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