Advanced Calculus Ch.0

Chapter 0. Preliminaries

Axiom. The Completeness Property of $\mathbb{R}$

\[\forall A \subset \mathbb{R}, A \neq \emptyset \Rightarrow \exists m \in \mathbb{R} \text{ s.t. } m = \sup A\]

Maximum and Minimum

Definition

\[\max(A) = m \Leftrightarrow m \in A \wedge \forall x \in A (x \leq m)\] \[\min(A) = m \Leftrightarrow m \in A \wedge \forall x \in A (x \geq m)\]

Property

\[\exists m \text{ s.t. the minimum or maximum of } E \Rightarrow m \text{ is unique.}\]

prove

$$ \text{Let } E \text{ be a nonempty subset of } \mathbb{R}. $$

Bounded Set

Definition

\[A \text{ is bounded above} \Leftrightarrow \exists M \in \mathbb{R} \text{ s.t. } \forall x \in A, x \leq M\] \[A \text{ is bounded below} \Leftrightarrow \exists m \in \mathbb{R} \text{ s.t. } \forall x \in A, x \geq m\] \[m \text{ is an upper bound of } A \Leftrightarrow \forall x \in A, x \leq m\] \[m \text{ is a lower bound of } A \Leftrightarrow \forall x \in A, x \geq m\]

Supremum and Infimum

Definition

\[\sup A = m \Leftrightarrow (\forall x \in A, x \leq m) \wedge (\forall y \in \mathcal{U}_A, y \geq m)\] \[\inf A = m \Leftrightarrow (\forall x \in A, x \geq m) \wedge (\forall y \in \mathcal{L}_A, y \leq m)\]

Property

\[\text{Let } \alpha \text{ be an upper bound of a subset } E \text{ of } \mathbb{R}. \text{ Then the following are equivalent:}\] \[\begin{align} & \sup E \text{ exists and } \alpha = \sup E \\ & \forall \gamma < \alpha, \exists x \in E \text{ s.t. } \gamma < x \leq \alpha \\ & \forall \epsilon > 0, \exists x \in E \text{ s.t. } \alpha - \epsilon < x \leq \alpha \end{align}\]

Reflection of a Set

Definition

\[-E = \{-x : x \in E\}\]

Property

\[\sup(E) = -\inf(-E)\]

Archimedean Principle

\[\forall a, b \in \mathbb{R}, a > 0 \Rightarrow \exists n \in \mathbb{N} \text{ such that } b < na\]

Well-Ordering Principle

Definition

\[\forall E \subset \mathbb{N}, E \neq \emptyset \Rightarrow \exists \min E\]

Extended Version

\[\forall X \subset \mathbb{Z}, X \neq \emptyset \Rightarrow \exists \max X\]

Dense Subset of $\mathbb{R}$

\[A \text{ is dense in } \mathbb{R} \Leftrightarrow \forall a, b \in \mathbb{R} \text{ with } a < b \Rightarrow \exists x \in A \text{ such that } a < x < b\]