Question

If \(a_{1}, a_{2}, \ldots, a_{n}\) are arbitrary real numbers, then $$ \left|a_{1}+a_{2}+\cdots+a_{n}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right| $$

Short answer

Question: Prove that for any arbitrary real numbers \(a_1, a_2, ..., a_n\), the absolute value of their sum is less than or equal to the sum of their absolute values. Answer: We prove this inequality using induction. For the base case when \(n=1\), the inequality holds as \(\left|a_1\right| \leq \left|a_1\right|\). Assuming the inequality holds for \(n=k\), we use the triangle inequality to show that it also holds for \(n=k+1\). Thus, the inequality holds for all positive integers n: \(\left|a_{1}+a_{2}+\cdots+a_{n}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right|\).

Step-by-step solution

Understand the given inequality

The inequality we need to prove is $$ \left|a_{1}+a_{2}+\cdots+a_{n}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right|. $$ This means that taking the absolute value of the sum of the arbitrary real numbers must be less than or equal to the sum of the absolute values of each of the numbers.

Employ induction to prove the inequality

To prove this inequality, we will use the principle of mathematical induction. First, we start with the base case when \(n=1\). Then, we will assume that the inequality holds for \(n=k\), and try to prove it for \(n=k+1\).

Base case when n=1

When \(n=1\), the inequality is simply $$ \left|a_1\right| \leq \left|a_1\right|. $$ This is true, proving the base case.

Inductive hypothesis

Now, let's assume that the inequality holds for \(n=k\): $$ \left|a_{1}+a_{2}+\cdots+a_{k}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{k}\right|. $$

Inductive step

We want to prove that the inequality holds for \(n=k+1\). So, we will consider the left side of the inequality: $$ \left|a_{1}+a_{2}+\cdots+a_{k}+a_{k+1}\right|. $$ By the triangle inequality, we know that \(\left| a+b\right| \leq \left| a \right| + \left| b \right|\) for any real numbers \(a\) and \(b\). Therefore, applying the triangle inequality to our problem, we have: $$ \left|a_{1}+a_{2}+\cdots+a_{k}+a_{k+1}\right| \leq \left|(a_{1}+a_{2}+\cdots+a_{k})\right|+\left|a_{k+1}\right|. $$ From our inductive hypothesis, we can replace the sum in the absolute value on the right side: $$ \left|a_{1}+a_{2}+\cdots+a_{k}+a_{k+1}\right| \leq \left(\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{k}\right|\right)+\left|a_{k+1}\right|. $$ Therefore, the inequality holds for \(n=k+1\), which completes the proof by induction: $$ \left|a_{1}+a_{2}+\cdots+a_{n}\right| \leq\left|a_{1}\right|+\left|a_{2}\right|+\cdots+\left|a_{n}\right|. $$

Key concepts

Triangle Inequality

The triangle inequality is a fundamental concept in mathematics. It deals with the relationship between the lengths of sides of a triangle, translated into inequalities for real numbers. When we talk about the triangle inequality in the context of real numbers, it states that for any real numbers \(a\) and \(b\), it holds that:

• \( \left| a + b \right| \leq \left| a \right| + \left| b \right| \).

This principle extends to objects like vectors and complex numbers, but even with real numbers, it is an essential tool.
In the step-by-step solution given, we apply the triangle inequality to a series of real numbers \(a_1, a_2, \ldots, a_n\). This rule helps us break down the sum of these numbers into more manageable parts, allowing us to compare the absolute value of a sum to the sum of absolute values.
Using the triangle inequality effectively helps to establish bounds and limits that are crucial in solving mathematical problems and proving statements within the realm of real analysis.

Absolute Value

The absolute value of a real number is a measure of its magnitude, without considering its direction. For example, both \(10\) and \(-10\) have an absolute value of \(10\). This is expressed mathematically as:

• \( \left| x \right| = x \) if \( x \geq 0 \)
• \( \left| x \right| = -x \) if \( x < 0 \)

The concept is widely used in mathematics to discuss distances and is vital in understanding the triangle inequality.
In the exercise, when we consider the sum of \(n\) real numbers, finding the absolute value of their sum helps us measure the total magnitude disregarding their positive or negative values. The inequality aims to show that this magnitude is never more than simply adding up the individual magnitudes (i.e., the absolute values).
Absolute value simplifies a wide array of computations because we do not need to worry about the sign of each element, only the size. It provides a consistent, non-negative quantity which is very useful in establishing comparisons and relationships.

Real Numbers

Real numbers are a class of numbers that include both rational numbers (such as \(3\), \(-1/2\), and \(7.5\)) and irrational numbers (such as \(\sqrt{2}\) and \(\pi\)). They encompass all the quantities that can be represented as a continuous number line. The properties of real numbers such as addition, subtraction, and multiplication are foundational in mathematics.
Understanding real numbers is crucial for tackling problems in calculus, algebra, and analysis. When the exercise mentions arbitrary real numbers \(a_1, a_2, \ldots, a_n\), it implies that these numbers can be any real values, possibly forming a mix of positives, negatives, or even zero.
The principle of mathematical induction used in the solution builds directly on the properties of real numbers. By proving the base case and using the inductive step over these numbers, the solution demonstrates that the inequality holds true regardless of the specific real numbers involved. This universality is one of the most powerful aspects of working with real numbers in mathematics.