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
- \(a^2+b^2 \geq 2ab\)
- \(\frac{a+b}{2} \geq \sqrt{ab}\)
Proof. \[ (\sqrt{a}-\sqrt{b})^2 = a - 2\sqrt{ab} + b \geq 0 \]
\[ \frac{a+b}{2} \geq \sqrt{ab} \]
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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
\[ \begin{align} \forall x \gt 0, \exists n \in \mathbb{N}, \frac{1}{n} \lt x \end{align} \]