Proofs Involving Real Numbers

Facts

• \(a^2 \geq 0, \forall a \in \mathbb{R}\)
• \(a^n \geq 0, \forall a \in \mathbb{R}\) if \(n \in \mathbb{Z}\) is even.
• \(a^n < 0, \forall a < 0\) if \(n \in \mathbb{Z}\) is odd.
• \([ab > 0] \iff a > 0\) INCOMPLETE

Let \(a, b, c \in \mathbb{R}\).

• \(a \geq b\), divide by \(c \geq 0\)
  • \(ac \geq bc\)
• \(a \geq b\), divide by \(c > 0\)
  • \(\tfrac{a}{c} \geq \tfrac{b}{c}\).
• \(a > b\), multiply or divide by \(c > 0\)
  • \(ac > bc\)
  • \(\tfrac{a}{c} > \tfrac{b}{c}\)
• \(a > b\), multiply or divide by \(c < 0\)
  • \(ac < bc\)
  • \(\tfrac{a}{c} < \tfrac{b}{c}\)

Theorem 4.13 (Zero Product Rule)

• Let \(x, y \in \mathbb{R}\).

• If \(xy = 0\), then \(x = 0\) or \(y = 0\).

  • Note: this provides a "proof technique"
    • If a product of reals is 0, then one of those reals must be 0.

• Proof: Assume \(xy = 0\).

  • WLOG, consider \(x\).
  • Case 1: \(x = 0\): Done :)
  • Case 2: \(x \neq 0\).
    • So \(\tfrac{1}{x}\) is defined.
    • Observe: \[ \begin{align} xy &= 0\\ \tfrac{1}{x}(xy) &= \tfrac{1}{x}(0)\\ y &= 0 \quad \blacksquare \end{align} \]

• Alternate Proof (Contrapositive)

  • Assume \(\neg(x = 0 \vee y = 0)\)
    • So \(x \neq 0 \wedge y \neq 0\)
  • From here, solve by cases (\(x > 0, y > 0\), etc)

Result 4.14: Let \(x \in \mathbb{R}\).

• If \(x^3-5x^2+3x = 15\), then \(x = 5\).
• Proof: Assume \(x^3 - 5x^2 + 3x = 15\).

\[ \begin{align} x^3 - 5x^2 + 3x - 15 &= 0\\ x^2(x-5) + 3(x-5) &= 0\\ (x^2+3)(x-5) &= 0\\ \end{align} \]

• By the Zero Product Rule, either \(x^2+3 = 0\) or \(x-5 = 0\).
  • \(x^2+3 = 0 \implies x^2=-3\) (contradiction)
  • \(x - 5 = 0 \implies x=5\) \(\blacksquare\)