Question

Use the Cauchy inequality to prove that: a. \(r_{1}+r_{2}+\cdots+r_{n} \leq n\left(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2}\right)\) for all \(r_{i}\) in \(\mathbb{R}\) and all \(n \geq 1\) b. \(r_{1} r_{2}+r_{1} r_{3}+r_{2} r_{3} \leq r_{1}^{2}+r_{2}^{2}+r_{3}^{2}\) for all \(r_{1}, r_{2},\) and \(r_{3}\) in \(\mathbb{R}\). [Hint: See part (a).]

Short answer

Both a and b are proved using the Cauchy-Schwarz inequality.

Step-by-step solution

Understand Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality states that for any real numbers or vectors in an inner product space, \( (a_1^2 + a_2^2 + ... + a_n^2)(b_1^2 + b_2^2 + ... + b_n^2) \geq (a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \). We will use this inequality to prove the given expressions.

Prove Part (a) using Cauchy-Schwarz

Apply the Cauchy-Schwarz inequality with \( a_i = 1 \) and \( b_i = r_i \). This gives \( (1^2 + 1^2 + \cdots + 1^2) \times (r_1^2 + r_2^2 + \cdots + r_n^2) \geq (r_1 + r_2 + \cdots + r_n)^2 \). Simplifying, we get \( n(r_1^2 + r_2^2 + \cdots + r_n^2) \geq (r_1 + r_2 + \cdots + r_n)^2 \). Taking the square root on both sides, since both sides are non-negative, \( r_1 + r_2 + \cdots + r_n \leq \sqrt{n} \times \sqrt{(r_1^2 + r_2^2 + \cdots + r_n^2)} \). Finally, this inequality simplifies back to the original form given, showing it holds true.

Apply Result of Part (a) to Prove Part (b)

For part (b), consider it as a specific case of part (a) with \( n = 3 \). The expression \( r_1r_2 + r_1r_3 + r_2r_3 \) can be viewed as \( \frac{1}{2}((r_1 + r_2 + r_3)^2 - (r_1^2 + r_2^2 + r_3^2)) \). Use the inequality proved in step 2. Thus, \( (r_1 + r_2 + r_3)^2 \leq 3(r_1^2 + r_2^2 + r_3^2) \). Therefore, \( r_1r_2 + r_1r_3 + r_2r_3 \leq \frac{1}{2}(3(r_1^2 + r_2^2 + r_3^2) - (r_1^2 + r_2^2 + r_3^2)) \). Simplifying yields \( r_1r_2 + r_1r_3 + r_2r_3 \leq r_1^2 + r_2^2 + r_3^2 \), which completes the proof for part (b).

Key concepts

Inequalities in mathematics

Inequalities in mathematics are statements that describe the relative size or order of two objects. They express that one quantity is larger or smaller than another. Let's look at different types of inequalities:

• **Strict Inequalities**: These are indicated by the symbols \(<, >\), where \(a < b\) means \(a\) is strictly less than \(b\), and \(a > b\) means \(a\) is strictly greater than \(b\).
• **Non-strict Inequalities**: These are indicated by \(\leq, \geq\), to express "less than or equal to" and "greater than or equal to".

The Cauchy-Schwarz inequality is an essential inequality in mathematics. It ensures that the magnitude of the dot product of two vectors is always less than or equal to the product of the magnitudes of the vectors. In simpler terms, for any real numbers or vectors in an inner product space:
\[(a_1^2 + a_2^2 + \ldots + a_n^2)(b_1^2 + b_2^2 + \ldots + b_n^2) \geq (a_1b_1 + a_2b_2 + \ldots + a_nb_n)^2\]This inequality is not just a cornerstone of mathematical theory but also has applications in fields such as physics and statistics.

Proof techniques in mathematics

Proof techniques in mathematics are methods used to show the truth of mathematical statements. These techniques are foundational tools in understanding and demonstrating mathematical concepts. Let's explore some common proof techniques:

• **Direct Proof**: Start from known facts and logical deductions to reach the statement that needs to be proven. This method was used in the Cauchy-Schwarz inequality by applying it directly to the given conditions.
• **Proof by Contradiction**: Assume the opposite of what you want to prove is true, and show this leads to a contradiction. This is useful in more complex scenarios.
• **Induction**: Used to prove statements about integers, typically by showing it's true for the first integer, and if true for \( n \), then true for \( n+1 \).

In our exercise with the Cauchy-Schwarz inequality, the direct proof method was effective. By substituting the inequality with specific values such as \(a_i = 1\) and \(b_i = r_i\), we derived a new inequality that confirmed the original problem statement. Using proof techniques correctly is crucial in ensuring statements are undeniably true.

Real numbers

Real numbers, denoted as \(\mathbb{R}\), form a fundamental building block in mathematics. They include all the numbers on the number line, encompassing:

• **Rational Numbers**: Fractions like \(\frac{1}{2}\) and whole numbers like \(3\). They can be written as the quotient of two integers.
• **Irrational Numbers**: Numbers that cannot be written as fractions, such as \(\sqrt{2}\) or \(\pi\). Their decimal expansions go on forever without repeating.

Real numbers are versatile in representing measurements and amounts. They are suitable for continuous data and are crucial in topics like calculus and geometry. In the context of our exercise, real numbers were used without restrictions on their values, focusing on their role in inequalities like the Cauchy-Schwarz inequality. Understanding real numbers is essential since they form the basis for more advanced mathematical concepts and applications.