Proof that the rationals are dense

• dalcde
In summary: Anyway, this is the way I see the proof.In summary, the conversation discusses a proof that the rationals are dense in the reals. The theorem states that for any two real numbers, there exists a rational number between them. However, the proof presented is incomplete and needs to address the concept of embedding the rationals in the reals. There is also a mistake in the proof where it is stated that there is a rational in x* but not in y*, when in fact every rational in x* is also in y*.
dalcde
Is the following proof that the rationals are dense in the reals valid?

Theorem: $$\forall x,y\in\mathbb{R}:x<y, \exists p\in\mathbb{Q}: x<p<y$$ Viewing x and y as Dedekind cuts (denoting the cuts as x* and y*), x* is a proper subset of y*, hence there exists a rational in x* but not in y*, i.e. there is a rational between x and y.

Last edited:
dalcde said:
Is the following proof that the rationals are dense in the reals valid?

Theorem: $$\forall x,y\in\mathbb{R}:x<y, \exists p\in\mathbb{Q}: x<p<y$$ Viewing x and y as Dedekind cuts (denoting the cuts as x* and y*), x* is a proper subset of y*, hence there exists a rational in x* but not in y*, i.e. there is a rational between x and y.

To prove the rationals are dense in the reals, you need to prove for any real number, there exists a rational number which is arbitrarily close to the real number. I don't see this in your proof.

Well, what do you mean by "arbitrarily close"? The usual definition of "there exist a rational number arbitrarily close to x" is "for any $\epsilon> 0$ there exist rational y such that $|x- y|< \epsilon$" which is the same as $-\epsilon< x- y< \epsilon$ or, in turn, $x-\epsilon< y< x+ \epsilon$ which is true if and only if "between any two real numbers there exist a rational number".

dalcde, when you say "x* is a proper subset of y*, hence there exists a rational in x* but not in y*" you have the inclusion wrong- there exist a rational in y* that is not in x*.

Also, if you are defining real number as Dedekind cuts (set of rational numbers), how are you embedding the rationals in the reals? The rational you are getting is a member of the set y*, not a real number itself.

(Yes, I know that you are associating the "rational cut", the set of all rational number less that r, with the rational number r. But you need to say that.)

Hi dalcde!

dalcde said:
Is the following proof that the rationals are dense in the reals valid?

Theorem: $$\forall x,y\in\mathbb{R}:x<y, \exists p\in\mathbb{Q}: x<p<y$$ Viewing x and y as Dedekind cuts (denoting the cuts as x* and y*), x* is a proper subset of y*, hence there exists a rational in x* but not in y*, i.e. there is a rational between x and y.

Your proof is not completed yet. You've proved that there is a rational in y*, but not in x*. So what will be the p* between x* and y* then? And why is p* not equal to x* and y*?

x* is the set of all rationals "below" x and y* is the set of all rationals "below" y. Hence there is some rational below x and not below y.

dalcde said:
x* is the set of all rationals "below" x and y* is the set of all rationals "below" y. Hence there is some rational below x and not below y.

I don't quite follow the "hence" bit. What exactly is the rational. And why doesn't it equal x and y?

dalcde said:
x* is the set of all rationals "below" x and y* is the set of all rationals "below" y. Hence there is some rational below x and not below y.

According to you, x < y. Let x* = {a in Q : a < x} and y* = {b in Q : b < y}. If z is in x*, then z < x < y. So z is in y*. So, you are wrong: every rational in x* is also in y*. HallsofIvy had already corrected you.

Sorry, I wanted to write x>y, and got it the wrong way round.

What does it mean for the rationals to be dense?

Being dense means that the rationals are closely packed together on the number line, with no gaps or jumps between them.

How is the density of the rationals proved?

The density of the rationals is proved by using the concept of limits. We can show that between any two rational numbers, there exists another rational number. This means that the rationals are infinite and closely packed together, fulfilling the definition of being dense.

What are some real-life examples of the density of the rationals?

One example is measuring a distance in meters. No matter how small the distance is, there will always be a rational number that represents it. This shows that the rationals are densely packed on the number line.

Can the density of the rationals be visualized?

Yes, the density of the rationals can be visualized on a number line. The closer the rational numbers are to each other, the denser they are. This can be seen by the smaller the distance between two rational numbers, the more numbers there are between them.

How does the density of the rationals compare to the density of the irrationals?

The density of the rationals is different from the density of the irrationals. While the rationals are densely packed together with no gaps, the irrationals have gaps between them on the number line. This is because the irrationals cannot be expressed as a ratio of two integers, so they are not included in the set of rational numbers.

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