Can You Prove There Are Infinite Rationals Between Two Real Numbers?

In summary, the conversation discusses a proof for the existence of at least one rational number between two arbitrary real numbers x and y, where x>y. The proof involves choosing \epsilon and n to satisfy the given conditions, and then using the fact that there exist infinite numbers for n to show that there are infinitely many rational numbers between x and y.
  • #1
Shing
144
1

Homework Statement


If x and y are arbitrary real numbers. x>y. prove that there exist at least one rational number r satisfying x<r<y, and hence infinitely.

The Attempt at a Solution


well, I have done my proof, but comparing to the solution offered by http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-014Calculus-with-Theory-IFall2002/1C8FA521-FDCE-491B-8689-955B04A4A4A2/0/pset2solutions.pdf" (*1), I have a bit doubt about whether my proof is precise enough or not.

anyway, here it is:

x,y belong to R, x<y
let[itex]|x-y|>\varepsilon[/itex]
let n belongs Z, n>1
obviously,[itex]\varepsilon[/itex] satisfies [itex]x<x+\frac{\varepsilon}{n}<y[/itex]
as there exist infinite numbers for n,
therefore, infinite r satisfy x<r<y

thanks for reading =)
 
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  • #2
Shing said:

Homework Statement


If x and y are arbitrary real numbers. x>y. prove that there exist at least one rational number r satisfying x<r<y, and hence infinitely.


The Attempt at a Solution


well, I have done my proof, but comparing to the solution offered by http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-014Calculus-with-Theory-IFall2002/1C8FA521-FDCE-491B-8689-955B04A4A4A2/0/pset2solutions.pdf" (*1), I have a bit doubt about whether my proof is precise enough or not.

anyway, here it is:

x,y belong to R, x<y
let[itex]|x-y|>\varepsilon[/itex]
let n belongs Z, n>1
obviously,[itex]\varepsilon[/itex] satisfies [itex]x<x+\frac{\varepsilon}{n}<y[/itex]
as there exist infinite numbers for n,
therefore, infinite r satisfy x<r<y

thanks for reading =)

If [itex]\epsilon[/itex] is not a rational number, then is [itex]\epsilon/n[/itex] rational?

If [itex]|x-y|>\epsilon[/itex] then can you find a rational number such that [itex]\epsilon[/itex] is larger than this rational number? The rest of your arguments can be used provided you find this rational number.
 
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FAQ: Can You Prove There Are Infinite Rationals Between Two Real Numbers?

What is the definition of a proof in real analysis?

A proof in real analysis is a rigorous demonstration that a mathematical statement or theorem is true using logical deductive reasoning and axioms. It involves breaking down a complex statement into smaller, more easily proven statements and using previously established theorems to arrive at a conclusion.

How do you know if a proof in real analysis is valid?

A valid proof in real analysis must follow the rules of logic and use accepted axioms and theorems. It should also be clear, concise, and free of contradictions. Additionally, the proof should be replicable by others, meaning that anyone who follows the same steps should arrive at the same conclusion.

What are some common techniques used in proofs in real analysis?

Some common techniques used in proofs in real analysis include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. In direct proof, the statement is proven by applying known definitions and properties. In proof by contradiction, the statement is shown to be true by assuming the opposite and arriving at a contradiction. In proof by induction, the statement is proven for a base case and then shown to hold for all subsequent cases. In proof by contrapositive, the statement is proven by showing that the negation of the statement is false.

Can you provide an example of a proof in real analysis?

One example of a proof in real analysis is the proof that the square root of 2 is an irrational number. This proof uses proof by contradiction and assumes the opposite, that the square root of 2 is a rational number. By expressing the square root of 2 as a fraction and simplifying, it can be shown that this leads to a contradiction, thus proving that the assumption was false and the square root of 2 is indeed irrational.

Why is it important to include proofs in real analysis?

Including proofs in real analysis is important because it provides a solid foundation for mathematical concepts and ensures that they are true and valid. Proofs also allow for a deeper understanding of mathematical concepts and can lead to the discovery of new theorems and properties. Additionally, proofs are essential in various fields such as physics, engineering, and computer science where mathematical principles are used to solve real-world problems.

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