Proof of Uniqueness of y for x > 0

[glebovg]

Homework Statement

If x > 0, then there exists a unique y > 0 such that y² = x.

The attempt at a solution

Proof. Let A = {y ∈ Q : y² < x}. A is bounded above by x, so lub(A) = η exists.
Suppose η² > x, where η = lub(A).
Consider (η - 1/n)² = η² - 2η/n +1/n² > η² - 2η/n.
Now η² - 2η/n > x ⇔ η² - x > 2η/n ⇔ (η² - x)/2η > 1/n.
We may choose such n by the Archmedean Property.
Thus η - 1/n is an upper bound and η = lub(A), a contradiction.
Similarly, if η² < x, consider (η + 1/n)² = η² + 2η/n +1/n² > η² + 2η/n.
Now η² + 2η/n < x ⇔ 2η/n < x - η² ⇔ 1/n < (x - η²)/2η.
We may choose such n. So η is not an upper bound.
Therefore, η² = x by the Trichotomy rule. ∎

 

[mighty2000]

Thank you for your post. I would like to provide some feedback on your attempt at proving the statement "If x > 0, then there exists a unique y > 0 such that y2 = x."

Firstly, I appreciate your use of notation and logical steps in your proof. However, there are a few areas where I believe you could improve and clarify your proof.

1. Define your variables: It would be helpful to define your variables at the beginning of the proof. For example, you could define A as the set of rational numbers y such that y2 is less than x. This would make it easier for the reader to understand your proof.

2. Use of the Archimedean Property: While your use of the Archimedean Property is correct, it would be helpful to explain why we can choose such an n. This property may not be familiar to all readers, so it would be helpful to provide a brief explanation or reference to its definition.

3. Clarify the contradiction: In your proof, you state that η = lub(A) is a contradiction, but it would be helpful to explain why this is a contradiction. One way to do this is to state that if η is the least upper bound of A, then there cannot be any number greater than η in A.

Overall, your proof is well-structured and logical. With a few clarifications and explanations, it would be a solid proof for the statement. Keep up the good work!