Question

In: Advanced Math

Recall that a set B is dense in R if an element of B can be...

Recall that a set B is dense in R if an element of B can be found between any two real numbers a < b. Take p∈Z and q∈N in every case. It is given that the set of all rational numbers p/q with 10|p| ≥ q is not dense in R. Explain, using plain words (without a rigorous proof), why this is. That is, present a general argument in plain words. Does this set violate the Archimedean Property? If so, how? (PLEASE DON'T REPEAT THE ANSWERS TO THIS QUESTION ALREADY POSTED ON CHEGG)

Solutions

Expert Solution

Sorry, I don't know about the  previous answers. I hope this one is clear to you.

Let

   .

If is dense in then by defenition, for ANY real number a and b, a<b, we can find an element in .

Now, we have to prove that is not dense in .

For that it is enough to prove that

.

  

----------------------------(i)

So let me choose  

Then clearly

  

which means if we choose ,

we cannot find an element of between .

ARCHIMEDEAN PROPERTY  

For any   and , there is an     so that which is equivalent to .

doesnot satisfies Archimedean property.

Because,   

   For any natural number  , there exists a positive real number  

where such that , which is a contradiction to Archimedean property.

  

Colby Messinger answered 1 year ago
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