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The concept of real numbers is fundamental to understanding the properties of absolute value in mathematics. Real numbers, denoted by \( \mathbb{R} \), include all rational and irrational numbers, which can be represented on the number line. They encompass whole numbers, integers, fractions, and decimals, both positive and negative, as well as the special number zero. What's important to note when dealing with real numbers is that operations on them obey certain rules and properties that enable us to perform addition, subtraction, multiplication, and division.

One such operation that is crucial in the study of real numbers in the context of absolute value is the multiplication of two real numbers. When two real numbers, \( a \) and \( b \) in our exercise, are multiplied, the sign of their product is determined by the signs of the multiplicands. For instance, a positive times a negative will always result in a negative product, while the product of two numbers with the same sign is always positive. This detail is pivotal to understanding the relationship between absolute values and the sign of the product of two numbers.

Understanding inequality relationships is key to grasping the more intricate parts of algebra, such as solving equations and inequalities that involve absolute values. An inequality indicates a comparative statement about the size of two quantities. For our case, we specifically look at the inequality \( ab \geq 0 \), which means that the product of \( a \) and \( b \) is either greater than or equal to zero. This is a crucial concept when looking at the absolute value of a sum of two real numbers.

The inequality \( ab \geq 0 \) implies that \( a \) and \( b \) are either both positive or both negative, or at least one of them is zero. These rules are what mathematicians and students use to deduce specific properties or conditions for given expressions. Indeed, in the context of the absolute value problem we're discussing, linking this inequality with the equality of the absolute values \( |a+b| \ = |a| + |b| \) helps to set the stage for a proof that explicates the conditions under which the equation holds true.

Proofs are the backbone of real analysis and give us the tools to verify the truth of mathematical statements rigorously. A proof is a logical argument that starts from known premises and uses deductive reasoning to arrive at a conclusion. In real analysis, we often deal with proofs related to the properties of real numbers and their functions.

In the case of our exercise, the proof revolves around showing the condition under which \( |a+b| \ = |a| + |b| \). The method used in this proof is a direct approach which begins by assuming the truth of the required equation and explores the implications to uncover the conditions needed for it to be true. By establishing a connection to the inequality relationship \( ab \geq 0 \) and understanding how the absolute value operates over the sum of two numbers, we complete the proof and provide a fundamental understanding of this absolute value property.

Proofs are not just about showing that a statement is true; they provide deep insights and help to solidify one's understanding of mathematical concepts. Therefore, when learning about proofs in real analysis, it is paramount to follow each logical step carefully and appreciate how the premises lead to the conclusion.