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 Real Numbers:If a and b are positive real numbers, thenthere exists a poitive interger n such that na > b.

Completeness Axiom: Each nonempty set ofreal numbers that 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 Real Numbers:If a and b are positive real numbers, thenthere exists a poitive interger n such that na > b.

Completeness Axiom: Each nonempty set ofreal numbers that 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

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

Now take a = here.    Then by the Archimedean propertythere exists a positive integer N such that Na >b    That isN > b.    So wehave > b/N.    This proves the result.

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.