Prove the following consequences of the order axioms.

- If and , then .
- If , then .
- If , then and are both both positive or and are both negative.
- If and , then .

* Proof. * First, by definition of we have

Then, by Axiom 8 (and since implies and implies ), we have

Thus, by Axiom 7 , and then using the field properties (Section I.3.3) we have

Hence, , i.e.,

* Proof. * First, implies , and then we have,

Thus, ; hence,

* Proof. * First, we cannot have or since by Theorem I.11 this would mean ; hence, we could not have .

Assume then that and . By assumption . Now, if is positive then

On the other hand, if then

Thus, implies as well. Hence, if is positive then so is , and if is negative then so is , i.e., either and are both positive or both negative

* Proof. * Since we have , and since we have . Then,

Hence,

*Related*