# Thread: Prove that the set of rational numbers that are not integers is dense in the reals

1. ## Prove that the set of rational numbers that are not integers is dense in the reals

This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in $\displaystyle \mathbb{R}$:

Prove that the set $\displaystyle \mathbb{Q}\backslash\mathbb{Z}$ of rational numbers that are not integers is dense in $\displaystyle \mathbb{R}$.

So far I have:

We know the rationals $\displaystyle \mathbb{Q}$ are dense in $\displaystyle \mathbb{R}$. In the case where there are no integers in the set $\displaystyle (a,b)$ where $\displaystyle a<b$, then $\displaystyle (a,b)\backslash\mathbb{N}=(a,b)$ and so $\displaystyle \mathbb{Q}\cap (a,b)\not=0$ as shown previously.

In the case where there is at least one integer $\displaystyle m$ in the set $\displaystyle (a,b)$, we want to show that there is also a non-integer in $\displaystyle (a,b)$.

I am not sure where to go from here, or if dividing it into cases is the correct approach. Could anyone give me some pointers? Thanks!

2. ## Re: Prove that the set of rational numbers that are not integers is dense in the real

Originally Posted by Ragnarok
This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in $\displaystyle \mathbb{R}$:

Prove that the set $\displaystyle \mathbb{Q}\backslash\mathbb{Z}$ of rational numbers that are not integers is dense in $\displaystyle \mathbb{R}$.

Suppose that $\displaystyle t\in \mathbb{Q}\setminus\mathbb{Z}$. Then using the floor function $\displaystyle t \notin \mathbb{Z}$ and so $\displaystyle \left\lfloor t \right\rfloor < t < \left\lfloor t \right\rfloor + 1$

If $\displaystyle \delta=\min\{t-\left\lfloor t \right\rfloor,\left\lfloor t \right\rfloor+1-t\}$ then $\displaystyle t\in O=\left( {t - \delta ,t + \delta } \right)\cap\mathbb{Z}=\emptyset.$

Now use the density of $\displaystyle \mathbb{Q}$ in $\displaystyle \mathbb{R}$ with respect to $\displaystyle O$.

3. ## Re: Prove that the set of rational numbers that are not integers is dense in the real

Originally Posted by Ragnarok
This is from Fitzpatrick's Advanced Calculus, where it has already been shown that the rationals are dense in $\displaystyle \mathbb{R}$:

Prove that the set $\displaystyle \mathbb{Q}\backslash\mathbb{Z}$ of rational numbers that are not integers is dense in $\displaystyle \mathbb{R}$.

To do it your way, this is an edit.
If $\displaystyle (a,b)\cap\mathbb{Z}=\emptyset$ then you are done.

If $\displaystyle (a,b)\cap\mathbb{Z}\ne\emptyset$ then let $\displaystyle j$ be the least integer in $\displaystyle (a,b)\cap\mathbb{Z}$.

Then If $\displaystyle (a,j)\cap\mathbb{Z}=\emptyset$ and $\displaystyle (a,j)\subset(a,b)$

4. ## Re: Prove that the set of rational numbers that are not integers is dense in the real

Thank you! I think I got it.

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### prove that the set of rational numbers is dense in r

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