Question

Exercise 2.4.3: Proving algebraic statements with direct proofs. Prove each of the following statements using a direct proof.

A. For any positive real numbers, x and y, x + y ≥ √xy

Answer 1

To prove the statement x + y ≥ √xy, we can use a direct proof by assuming that x and y are positive real numbers. By expanding and simplifying, we can show that (x + y)² - 4xy simplifies to (x - y)² ≥ 0. Adding 4xy to both sides, we can then deduce that (x + y)² ≥ 4xy. By taking the square root and simplifying, we can conclude that x + y ≥ √xy.

Step-by-step explanation:

To prove the statement x + y ≥ √xy for any positive real numbers x and y, we can use a direct proof.

1. Assume that x and y are positive real numbers.
2. Consider the expression (x + y)² - 4xy. Expand and simplify it to obtain (x - y)² ≥ 0.
3. Since (x - y)² ≥ 0, this implies that x² - 2xy + y² ≥ 0.
4. Adding 4xy to both sides of the inequality gives x² + 2xy + y² ≥ 4xy.
5. Factoring the left side gives (x + y)² ≥ 4xy.
6. Taking the square root of both sides, we have x + y ≥ √(4xy).
7. Simplifying further, we get x + y ≥ 2√xy.
8. Since √(4xy) = 2√xy, we can conclude that x + y ≥ √xy.

Therefore, we have proved that for any positive real numbers x and y, x + y ≥ √xy using a direct proof.