Direct Proof Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Prove directly: if x>y>0x > y > 0, then x2>y2x^2 > y^2.

Solution

  1. 1
    Assume x>y>0x > y > 0.
  2. 2
    Since x>yx > y and both are positive, multiply both sides of x>yx > y by xx (positive): x2>xyx^2 > xy.
  3. 3
    Similarly, multiply x>yx > y by yy (positive): xy>y2xy > y^2.
  4. 4
    By transitivity: x2>xy>y2x^2 > xy > y^2, so x2>y2x^2 > y^2. □\square

Answer

x2>y2□x^2 > y^2 \quad \square
The key insight is to break x2>y2x^2 > y^2 into two steps via the intermediate term xyxy. Each step multiplies an inequality by a positive quantity, which preserves the direction of the inequality.

About Direct Proof

A direct proof establishes a statement P⇒QP \Rightarrow Q by assuming PP is true and using logical steps, definitions, and known theorems to arrive at QQ — the most straightforward proof strategy.

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More Direct Proof Examples

Example 1 easy

Prove directly: the sum of an even integer and an odd integer is odd.

Example 2 medium

Prove directly: for all real [formula], [formula].

Example 4 medium

Prove directly: for all integers [formula], [formula] is even.

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