# Show that if a, b∈ R with a < b, then there exists a number x ∈ I with a < x < b

• cooljosh2k2
In summary, using the density theorem, we can find a rational number between a/√2 and b/√2, which can be multiplied by √2 to obtain an irrational number x satisfying a < x < b. This proof may be supplemented with a proof of r√2 not being rational, unless it has already been shown in class or referenced in notes.
cooljosh2k2

## Homework Statement

Let I be the set of real numbers that are not rational. (We call
elements of I irrational numbers. Show that if a, b are real
numbers with a < b, then there exists a number x ∈ I with
a < x < b.

## The Attempt at a Solution

Im not sure if this is right:

Using the density theorem to the real numbers a/√2 and b/√2, we obtain a rational number such that:

a/√2 < r < b/√2

where r ≠ 0

This gives ---> x = r√2, which is an irrational number because it contains √2, which itself is irrational. This satisfies b < x < a

Is this right? or do i need more in the proof?

Perfect.

For the sake of completeness, you might want to add a proof of r \sqrt{2} not being rational. However, if you have shown this result in class or in your notes you should probably reference it.

## 1. What does "a < b" mean in this statement?

"a < b" means that a is less than b. In other words, a is a smaller number than b on the number line.

## 2. What is R in this statement?

R stands for the set of real numbers. This includes all numbers that can be represented on a number line, including positive and negative numbers, fractions, and decimals.

## 3. What does "x ∈ I" mean?

"x ∈ I" means that x is an element of the interval I. In other words, x is a number that falls between the values of a and b on the number line.

## 4. Why is it important to show that this statement is true?

This statement is important because it proves the existence of a number x that falls between any two real numbers a and b. This is a fundamental concept in mathematics and is used in various applications such as calculus and geometry.

## 5. Can you provide an example of a and b where this statement is true?

Yes, for example, if a = 1 and b = 2, then there exists a number x between 1 and 2, such as 1.5, that satisfies the statement. This can be visualized on a number line where a is to the left of b and x is a point between them.

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