Proof of Infinite Set of Real Numbers Between Two Unequal Real Numbers

[RandomAllTime]

Hello everybody, I seek the analysis and criticism of mathematicians. I'm sure this has already been proven a while ago, but I wrote a proof that there is an infinite set of real numbers between any two real numbers provided they are unequal. I am not yet in college and I lack proper training in proof writing, so I come here to show you the proof and gather whether or not you think it is valid, or if it is broken and or needs revision. Thank you. Perhaps even my assumption is wrong. I am just a beginner who is trying to develop proof writing skills, so please excuse any trivial errors, but please do tell of them. Here is what I have come up with. It is quite short and simple.
Consider the closed interval [a,b], where a≠b and a,b ∈ ℝ.
Next consider the infinite sequence S = [/0], [/1], [/2], ... such that [/1] > a and limS = b.
The set of all terms in S shall be denoted X = {s|s∈S}
Since X is a subset of all the real numbers between a and b, and X is infinite, the set between any two real
numbers a and b is infinite.

Thank you.

 

[HallsofIvy]

That's very difficult to understand. Apparently you are trying to prove that there are an infinite number of real numbers between any two numbers, a and b. You start by saying "consider the infinite sequence S = [/0], [/1], [/2], ... such that [/1] > a and limS = b." and then conclude that the infinite set of numbers in that sequence are in the interval [a, b]. That is NOT true because you do NOT require that the numbers in the sequence be less than b.

For example, taking a= 1, b= 2, and the sequence to be 2+ 1/(n+1) satisfies all of conditions you give but none of the numbers in the sequence are in [1, 2]. You are assuming, without saying it, that your sequence lies in the interval [a, b] but that assumes what you purport to prove.

 

[RandomAllTime]

I'm sorry. I screwed up with the subscript notation (I'm new to it). I was trying to convey that S={s(subscript 0),s(subscript 1), ...} and that s(subscript 0) > b. I imagine my posted version looked quite ridiculous. Thank you.

 

[jbriggs444]

I'm sorry. I screwed up with the subscript notation (I'm new to it). I was trying to convey that S={s(subscript 0),s(subscript 1), ...} and that s(subscript 0) > b. I imagine my posted version looked quite ridiculous. Thank you.

There are a couple of rules of mathematical discourse that sometimes go unstated:

If you say "a <something>" then you have to be able to produce a proof of existence. You have to be able to demonstrate that at least one such something exists.

If you say "the <something>" then you have to be able to produce a proof of existence and uniqueness. You have to be able to demonstrate that at least one such something exists. And you have to be able to show that there is only one.

In the original post above you refer to "the infinite sequence S...". But no proof of either existence or uniqueness is provided.

 

[geoffrey159]

Hmmm, not the funniest thing to do during summer time

This is where you should start : https://en.wikipedia.org/wiki/Archimedean_property

Then you will try to prove that between any 2 real number, there is a rational number. By induction, you will deduce that there is an infinite sequence of rational number between 2 real number, thus proving your property. This famous property is called "density of $\mathbb{Q}$ in $\mathbb{R}$".

 

[RandomAllTime]

There are a couple of rules of mathematical discourse that sometimes go unstated:

If you say "a <something>" then you have to be able to produce a proof of existence. You have to be able to demonstrate that at least one such something exists.

If you say "the <something>" then you have to be able to produce a proof of existence and uniqueness. You have to be able to demonstrate that at least one such something exists. And you have to be able to show that there is only one.

In the original post above you refer to "the infinite sequence S...". But no proof of either existence or uniqueness is provided.

There are a couple of rules of mathematical discourse that sometimes go unstated:

If you say "a <something>" then you have to be able to produce a proof of existence. You have to be able to demonstrate that at least one such something exists.

If you say "the <something>" then you have to be able to produce a proof of existence and uniqueness. You have to be able to demonstrate that at least one such something exists. And you have to be able to show that there is only one.

In the original post above you refer to "the infinite sequence S...". But no proof of either existence or uniqueness is provided.

Ok, thank you very much.

 

[RandomAllTime]

Hmmm, not the funniest thing to do during summer time

This is where you should start : https://en.wikipedia.org/wiki/Archimedean_property

Then you will try to prove that between any 2 real number, there is a rational number. By induction, you will deduce that there is an infinite sequence of rational number between 2 real number, thus proving your property. This famous property is called "density of $\mathbb{Q}$ in $\mathbb{R}$".

Ah, I see. Thank you :)

 

[RandomAllTime]

Thank you all for the feedback, and please excuse my messed up notation in the original post.

 

[micromass]

Hmmm, not the funniest thing to do during summer time

This is where you should start : https://en.wikipedia.org/wiki/Archimedean_property

Then you will try to prove that between any 2 real number, there is a rational number. By induction, you will deduce that there is an infinite sequence of rational number between 2 real number, thus proving your property. This famous property is called "density of $\mathbb{Q}$ in $\mathbb{R}$".

The Archimedean property is definitely overkill here since the same result is true for non-Archimdean fields. In fact, an easy proof just uses the sequence of successive averages:

a_0 = a,~a_1 = b, ~a_2 = \frac{a + b}{2},..., a_n = \frac{a_{n-1} + a_{n-2}}{2},...

 

[aikismos]

Have to agree with @micromass.

The infinite density of the reals in more detail: http://www.abstractmath.org/MM/MMRealDensity.htm

 

[geoffrey159]

The Archimedean property is definitely overkill here since the same result is true for non-Archimdean fields. In fact, an easy proof just uses the sequence of successive averages:

a_0 = a,~a_1 = b, ~a_2 = \frac{a + b}{2},..., a_n = \frac{a_{n-1} + a_{n-2}}{2},...

definitely agree with you

 

[DrewD]

Thank you all for the feedback, and please excuse my messed up notation in the original post.

If you are really interested in this stuff (math I guess?) it is a good idea to learn LaTeX. It will help you on this site and it will help you in college. If you are still in HS, it isn't really important, but it also isn't really that hard to pick up the basics.

 

[berkeman]

If you are really interested in this stuff (math I guess?) it is a good idea to learn LaTeX.

And here is a tutorial thread to help you get started:

https://www.physicsforums.com/threads/latex-faq.617570/

 

[RandomAllTime]

If you are really interested in this stuff (math I guess?) it is a good idea to learn LaTeX. It will help you on this site and it will help you in college. If you are still in HS, it isn't really important, but it also isn't really that hard to pick up the basics.

Thanks, will do.

 

[RandomAllTime]

And here is a tutorial thread to help you get started:

https://www.physicsforums.com/threads/latex-faq.617570/

Thanks bud!

 

[RandomAllTime]

Have to agree with @micromass.

The infinite density of the reals in more detail: http://www.abstractmath.org/MM/MMRealDensity.htm

Thanks, very interesting read.

 

[RandomAllTime]

Thanks, very interesting read.

Huh, maybe I'm wrong, but it's interesting to think of ℝ as a sort of fractal object. I mean, as you zoom in there will be infinite line division. The vertical length of line division n would be equal to |n|. The integers can be the taller line divisions while the non integer reals are smaller and make up the infinite space inbetween them. Also, maybe the different types of line divisions could have a different color scheme, like the primes be blue, and whatnot. The point zero would be like the middle between these two "elephant ear" type shapes which would be the negative and positive sides. Just an abstraction I find interesting. What do you guys think?

 

[aikismos]

Huh, maybe I'm wrong, but it's interesting to think of ℝ as a sort of fractal object. I mean, as you zoom in there will be infinite line division. The vertical length of line division n would be equal to |n|. The integers can be the taller line divisions while the non integer reals are smaller and make up the infinite space inbetween them. Also, maybe the different types of line divisions could have a different color scheme, like the primes be blue, and whatnot. The point zero would be like the middle between these two "elephant ear" type shapes which would be the negative and positive sides. Just an abstraction I find interesting. What do you guys think?

I think that you have excellent intuition, and that the intuition is anticipating that that fractals are a mathematical construct that build upon the infinite density property of the reals. Hmmmm. Certainly what most of us visualize when we hear the world 'fractals' (see https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) is topological in nature and requires a mapping of the Reals onto a space. I think the Extended Real Number Line fails to be a fractal because of the requirement that by definition a fractal (L. fractus, broken) has to have a non-integer dimension. The ERNL by definition has a dimension of 1. If you look at the list to which I've linked above, you'll see that the Cantor set (which is a subset of the ERNL) is a subset of the Reals which does qualify.

You seem to be genuinely interested in mathematics for its own sake. How motivated are you to learn and do math? I ask because I'm an ex-teacher (computer science and math), and I wanted to undertake a bit of a math project online and would be interested in your curiosity and skills. If you're really like me (I can never get enough of math), it might be some good experience for you.

 

[RandomAllTime]

I think that you have excellent intuition, and that the intuition is anticipating that that fractals are a mathematical construct that build upon the infinite density property of the reals. Hmmmm. Certainly what most of us visualize when we hear the world 'fractals' (see https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) is topological in nature and requires a mapping of the Reals onto a space. I think the Extended Real Number Line fails to be a fractal because of the requirement that by definition a fractal (L. fractus, broken) has to have a non-integer dimension. The ERNL by definition has a dimension of 1. If you look at the list to which I've linked above, you'll see that the Cantor set (which is a subset of the ERNL) is a subset of the Reals which does qualify.

You seem to be genuinely interested in mathematics for its own sake. How motivated are you to learn and do math? I ask because I'm an ex-teacher (computer science and math), and I wanted to undertake a bit of a math project online and would be interested in your curiosity and skills. If you're really like me (I can never get enough of math), it might be some good experience for you.

Thank you for the compliment and the wonderful input! I'll be sure to check that out. I would also like to hear about your project.