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)

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

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