Semester 1 / Mathematics / Real Analysis / Order Axioms

Order Axioms

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  • Trichotomy
    ∀a,b∈R\forall a,b \in \mathbb{R} exactly one of these holds: a>ba > b, a=ba = b, a<ba<b
  • Transitivity
    ∀a,b,c∈R;a<b∧b<c  ⟹  a<c\forall a,b,c \in \mathbb{R}; a<b \land b<c \implies a<c
  • Operation with addition
    ∀a,b∈R;a<b  ⟹  a+c<b+c\forall a,b \in \mathbb{R}; a<b \implies a+c<b+c
  • Operation with mutliplication
    ∀a,b,c∈R;a<b∧0<c  ⟹  ac<bc\forall a,b,c \in \mathbb{R}; a<b \land 0<c \implies ac<bc

Definitions

  • a<b≡b>aa < b \equiv b > a
  • a≤b≡a<b∨a=b a \le b \equiv a<b\lor a=b
  • a≠b≡a<b∨a>b a\not =b\equiv a<b\lor a>b
  • ∣x∣={xif x≥0,−xif x<0\lvert{x}\rvert=\begin{cases}x & \text{if } x \ge 0, \\-x & \text{if } x < 0\end{cases}

Triangular Inequalities

∣a∣−∣b∣≤∣a+b∣≤∣a∣+∣b∣∣∣a∣−∣b∣∣≤∣a+b∣\begin{aligned} \lvert{a}\rvert -\lvert{b}\rvert & \le \lvert{a+b}\rvert \le \lvert{a}\rvert+ \lvert{b}\rvert \\ \Big\lvert{ \lvert{a}\rvert -\lvert{b}\rvert } \Big\rvert & \le \lvert{a+b}\rvert \end{aligned}

Required Proofs

  • ∀a,b,c∈R;a<b∧c<0  ⟹  ac>bc\forall a,b,c \in \mathbb{R}; a<b \land c<0 \implies ac>bc
  • 1>01 > 0
  • −∣a∣≤a≤∣a∣-\lvert{a}\rvert\le a\le\lvert{a}\rvert
  • Triangular inequalities

Epsilon Bounds

  • ∃a  ∀ϵ>0, a<ϵ  ⟹  a≤0\exists{a}\;\forall{\epsilon>0},\,a<\epsilon\implies{a}\le{0}
  • ∃a  ∀ϵ>0, 0≤a<ϵ  ⟹  a=0\exists{a}\;\forall{\epsilon>0},\,0\le{a}<\epsilon\implies{a}={0}
  • ∀ϵ>0  ∃a,a<ϵ\forall\epsilon>0\;\exists a,a<\epsilon does not imply a≤0a\le 0
Written by October 19, 2026 1 min read
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