Complete the following alternative proof of the theorem in...

Complete the following alternative proof of the theorem in Example 3.1.2.
Proof. Suppose $0<a<b$. Then $b-a>0$.
[Fill in a proof of $b^2-a^2>0$ here.]
Since $b^2-a^2>0$, it follows that $a^2<b^2$. Therefore if $0<a<b$ then $a^2<b^2$.

Key Concepts

Difference of Squares Factorization

This concept involves the algebraic identity that any two squared terms can be written as the product of their sum and difference, specifically, b² - a² = (b - a)(b + a). This factorization is fundamental in algebra and is a tool for simplifying expressions and proving inequalities, as it breaks a seemingly complex difference of squares into simpler multiplicative components.

Properties of Inequalities in the Real Numbers

In dealing with inequalities, a crucial idea is that if two numbers are positive, then their product must also be positive. This principle is used in proofs where demonstrating that the product of two factors is positive leads directly to the conclusion that one quantity is less than another. It relies on the ordered field axioms of the real numbers, which guarantee that the ordering is preserved under multiplication when all factors involved are positive.

Ordered Field Axioms

These are the fundamental properties of the real numbers, including the existence of additive and multiplicative identities and inverses, the distributive property, and importantly the order properties. The order properties ensure that if a and b are real numbers such that 0 < a < b, then any arithmetic operations—like addition or multiplication with positive numbers—will preserve this order. They underpin the logic behind proofs of inequalities, such as showing that a² < b² when 0 < a < b.

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