(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let x, y, and z be real numbers. Prove the following:

1. If x * z = y * z, then x = y.

2. If x is not equal to 0, then x^2 > 0. (consider the two cases x > 0 and x < 0 ).

3. 0 < 1

4. For each n ∈ N, if 0 < x < y, then x^n < y^n

5. If x > 1, then x^2 > x.

6. If 0 < x < 1, then x^2 < 1

7. If 0 < x < y, then 0 < √x < √y

8. If x > 0, then 1/x > 0. If x < 0, then 1/x < 0.

9. If 0 < x < y, then 0 < 1/y < 1/x.

10. If xy > 0, then either (i) x > 0 and y > 0, or (ii) x < 0 and y < 0.

2. Relevant equations

Ordered Field Axioms

3. The attempt at a solution

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# Difficult Analysis proofs

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