How to Prove Existence of Integer n in Dedekind Cut Using Archimedean Property?

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In summary, to prove that there exists an integer n such that nw is a member of α and (n+1)w is not a member of α, we can use the Archimedean property of Q. By finding an n such that nw < p < (n+1)w for any member p of α, we can show that nw is a member of α. If there is no such n, then nw is a member of α for every n by induction. However, since α is not equal to Q and Q is Archimedean, this is not possible. This means that if q is not a member of α, then q is greater than any member of α, and thus nw is also greater than q
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Shaji D R
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Let α be a Dedekind Cut. w a positive rational.How to prove that there exists a integer n such that nw is a member of α and (n+1)w is not a member of α, using Archemedian propoerty of Q.

Suppose p is a member of α. we can find n such that nw < p < (n+1)w. So nw is
a member of α. Further I am not able to proceed.

Please help me.
 
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OK. Looks like I got the answer. It was a trivial case. If there is no such n then nw is a member of alpha for every n by induction. Since aplpha is not equal to Q and Q is archemedian it is not possible.If q is not a member
of alpha then q > any member of alpha and nw > q for some n.And nw cannot be a member of alpha.

Sporry for asking such a simple question.
 

FAQ: How to Prove Existence of Integer n in Dedekind Cut Using Archimedean Property?

1. What is the Dedekind Cut?

The Dedekind Cut is a mathematical concept introduced by the German mathematician Richard Dedekind. It is a way to construct the real numbers from rational numbers by dividing the rational numbers into two sets based on a specific dividing point, also known as the Dedekind cut. This dividing point represents an irrational number and allows for the creation of an uncountable set of real numbers.

2. What is an Archimedean Q?

An Archimedean Q is a type of ordered field that satisfies the Archimedean property. This property states that for any two positive elements in the field, there exists a positive integer that is larger than the product of these two elements. In other words, the field does not contain any infinitely large or infinitely small elements.

3. How are Dedekind Cuts related to Archimedean Q?

Dedekind Cuts and Archimedean Q are both concepts in mathematics that involve the construction of the real numbers. Dedekind Cuts use the concept of dividing rational numbers to create real numbers, while Archimedean Q uses the Archimedean property to define an ordered field that contains real numbers.

4. What is the importance of Dedekind Cuts and Archimedean Q?

Dedekind Cuts and Archimedean Q are important concepts in mathematics because they provide a foundation for the construction of the real numbers, which are essential in many mathematical fields. They also help to bridge the gap between rational and irrational numbers, allowing for a better understanding of the continuum of numbers.

5. How do Dedekind Cuts and Archimedean Q contribute to mathematical analysis?

In mathematical analysis, Dedekind Cuts and Archimedean Q are used to define and prove many important theorems and concepts. For example, they are used in the construction of the real numbers and in the proof of the intermediate value theorem. They also play a role in the study of limits, continuity, and differentiability of functions.

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