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# Sum of negatives is negative.

Prove that the sum of two negative numbers is negative.

Proof. Let be negative numbers, i.e., and . By Theorem I.25, . Thus, is negative as well .

# Prove some consequences of the order axioms

Prove the following consequences of the order axioms.

1. If and , then .
2. If , then .
3. If , then and are both both positive or and are both negative.
4. If and , then .

1. 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., 2. Proof. First, implies , and then we have, Thus, ; hence, 3. 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 4. Proof. Since we have , and since we have . Then, Hence, 