Problem 16 - Book of Proof

Question

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) If \(a\) and \(b\) are positive real numbers, then \(a+b \geq 2 \sqrt{a b}\).

Step-by-Step Solution

Answer

Our proof by contradiction has shown that for any positive real numbers \(a\) and \(b\), the statement \(a + b \geq 2\sqrt{ab}\) is always true.

Step 1: State the Contrary Assumption

Assume to the contrary, that there exist positive real numbers \(a\) and \(b\) such that \(a+b < 2 \sqrt{a b}\). This is our contrary assumption.

Step 2: Square Both Sides

Since both sides of the inequality are positive, we can square both sides without changing the direction of the inequality. It results in \(a^2 + 2ab + b^2 < 4ab\).

Step 3: Rearrange the Terms

Now, let's rearrange the terms to get: \(a^2 - 2ab + b^2 < 0\). This can be simplified as: \((a-b)^2 < 0\).

Step 4: Find Contradiction

However, the square of any real number (including \(a-b\)) is always greater or equal to zero. This is a contradiction because we assumed at the beginning that \((a-b)^2\) is less than zero.

Step 5: Conclusion of Proof

Since we have reached a contradiction, our contrary assumption must be false. Therefore, for all positive real numbers \(a\) and \(b\), it must be true that \(a + b \geq 2\sqrt{ab}\). This completes the proof by contradiction.

Key Concepts

Inequality Proof

Inequality proofs are a crucial part of mathematics, helping us establish the limits or bounds of mathematical expressions. In this exercise, we are proving an inequality involving real numbers: if both numbers are positive, then the sum of two numbers is always at least twice the square root of their product.
To delve into this, we compare two sides of an inequality to show that one side is always larger or equal to the other.
This particular proof by contradiction attempts to show that assuming otherwise leads to a logical inconsistency.

• The inequality statement here is \( a + b \geq 2 \sqrt{ab} \).
• We use properties of real numbers to manipulate and test this inequality.
• Understanding this helps us see how inequalities describe relationships between quantities.

Conquering inequality proofs opens doors to understanding advanced mathematical concepts and their applications.

Mathematical Proof Methods

In mathematics, proof methods are the tools we use to establish the truth of statements. Different methods suit different types of problems.
Three prominent methods are direct proof, contrapositive proof, and proof by contradiction.

• Direct Proof: This method straightforwardly establishes the truth of a statement by logical reasoning from known facts.
• Contrapositive Proof: It involves proving the contrapositive of a statement, which is logically equivalent to the statement.
• Proof by Contradiction: This method assumes the opposite of what we want to prove and then shows this assumption leads to a contradiction.

The exercise uses proof by contradiction, showing that while the opposite assumption of the statement is made, it fails logically. These proof methods are foundations for deeper mathematical reasoning.

Real Numbers

Real numbers make up the set of numbers used most commonly in everyday life, including both rational and irrational numbers. They cover all the points on the number line, making them essential in mathematics.
Understanding real numbers within a proof context involves recognizing properties such as positivity, ordering, and squaring.

• Positive Real Numbers: Numbers greater than zero and play a central role in proving the inequality discussed.
• Ordering: Real numbers can be compared, facilitating the logic flow in inequality proofs.
• Squaring: Helps manipulate expressions to reveal relationships within a proof.

Grasping these properties provides insight into how real numbers behave under various mathematical operations and proofs.

Contradiction

Proof by contradiction is an elegant method that highlights the inconsistencies in false assumptions. This method is used when proving statements directly could be complex.
We begin by assuming the opposite of what we aim to prove; this is known as the "contrary assumption." For example, in the exercise, the contrary is assuming \( a+b < 2 \sqrt{ab} \).

• The process involves logical deductions from this assumption, scrutinizing it under mathematical laws.
• When the assumption leads to an absurdity or falsehood (like squaring both sides and ending with a contradiction), it implies the original statement must be true.

Such a method not only verifies results but strengthens the understanding of why a statement is inherently valid.