Inequalities

Math
Inequality
Published

January 22, 2025

TODO: inequality

Triangle Inequality

The sum of any two sides > the remaining side

Theorem 1 (Triangle Inequality) \[ |a + b| \leq |a| + |b| \]

Corollary 1  

  • \(||a| - |b| |\leq |a-b|\)
  • \(|a-b| \leq |a| + |b|\)

Proof. \[ \begin{align*} &|a| = |a-b+b| \leq |a-b|+|b| \\ &|a| -|b| \leq |a-b| \\ \end{align*} \]

Remark 1. \[ \begin{align*} &|b| = |b-a+a| \leq |b-a|+|a| \\ &|b|-|a| \leq |b-a| \\ &|a|-|b| \geq -|b-a| \\ \end{align*} \]

Theorem 2  

  1. \(a^2+b^2 \geq 2ab\)
  2. \(\frac{a+b}{2} \geq \sqrt{ab}\)

Proof. \[ (\sqrt{a}-\sqrt{b})^2 = a - 2\sqrt{ab} + b \geq 0 \]

\[ \frac{a+b}{2} \geq \sqrt{ab} \]

[^1]

2. Bernoulli’s Inequality

\[ \begin{align} \forall n \in \mathbb{N} , (1+x)^n \leq 1+nx \end{align} \]

3. Archimedean Property

There not exists a infinitely large real number

\[ \begin{align} \forall x \in \mathbb{R}, \exists n \in \mathbb{N}, x \lt n \end{align} \]

There not exists a infinitely small positive real number

\[ \begin{align} \forall x \gt 0, \exists n \in \mathbb{N}, \frac{1}{n} \lt x \end{align} \]