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Real Analysis is a branch of mathematics that deals with real numbers and the real-valued functions of a real variable. At its core, real analysis involves concepts like limits, continuity, integration, differentiation, and sequences. The study of inequalities is a significant part of real analysis, as it provides essential insights into the behavior of functions and sequences.

In our exercise, we're dealing with the real numbers 'a' and 'b', and we're asked to prove an inequality involving these two positive numbers. The foundation of real analysis allows us to use properties of real numbers to manage and manipulate inequalities and understand the interplay between different numbers and expressions. For instance, knowing that if 'a' is less than 'b', then 'a' squared will also be less than 'a' times 'b', uses the ordering of real numbers to make logical deductions about their behaviors.

Mathematical inequalities are expressions that describe the relative size or order of two values. They are pivotal tools in real analysis, playing a key role in establishing bounds and limits of functions and sequences. The problem at hand requires understanding how multiplication of inequalities operates. When the inequality is between two positive numbers, such as in the case where '0 ≤ a < b', multiplying both sides by a positive number preserves the inequality.

This rule is crucial when we multiply both sides of 'a < b' by 'a' to get 'a² < ab', knowing that 'a' is positive based on the given condition. The concept of mathematical inequalities extends beyond this, covering various types of inequalities like linear, polynomial, and rational, each with their own set of rules for manipulation and solving.

Proof techniques are the various methods used by mathematicians to establish the truth of mathematical statements. These techniques often include direct proof, proof by contradiction, induction, and counterexample among others. In this exercise, we utilized a direct proof to show the validity of the inequality 'a² ≤ ab < b²'.

Using direct proof, we took the initial given inequality, applied algebraic operations such as multiplication and division, and logically deduced the result without assuming the contrary. Additionally, we showed that the strict inequality 'a² < ab < b²' does not hold by providing a counterexample. This illustrates that when proving mathematical statements, it's equally important to test for the boundaries of the proposition, such as when the equality holds, which in this case is when 'a = 0' and 'b = 1'.