Introduction to Real Analysis - Robert G. Bartle | Donald R. Sherbert
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Chapter 1 Preliminaries
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1.1 Sets and Functions
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- If an element $x$ is in a set $A$, we write $$x\in A$$ and say that $x$ is a **member** of $A$, or that $x$ **belongs** to $A$. If $x$ is *not* in $A$, we write $$x\not\in A.$$

- If every element of a set $A$ also belongs to a set $B$, we say that $A$ is a **subset** of $B$ and write $$A \subseteq B \text{ or } B \supseteq A.$$

- A set $A$ is a **proper subset** if $A \subseteq B$, but there is at least *one element* of $B$ that is not in $A$. We write $$A \subset B$$

> **Definition 1.1.1** Two sets $A$ and $B$ are said to be **equal**, and we write $A=B$, if they contain the same elements.

- To prove that the sets $A$ and $B$ are equal, we must show that $$ A \subseteq B \text{ and } B \supseteq A.$$

- The set of **natural numbers** $\mathbb{N}$ $:=$ {$1, 2, 3, ...$},
- The set of **integers** $\mathbb{Z}$ $:=$ {$-2, -1, 0, 1, 2, ...$},
- The set of **rational numbers** $\mathbb{Q}$ $:=$ {$\frac{m}{n}$: $m, n \in$ $\mathbb{Z}$ and $n \neq 0$ },
- The set of **real numbers** $\mathbb{R}$.

>**Definition 1.1.3**
>(i) The **union** of sets $A$ and $B$ is the set $A \cup B :=$ {$x$ : $x \in A$ or $x \in B$ }.
>(ii) The **intersection** of the sets $A$ and $B$ is the set $A \cap B :=$ {$x$ : $x \in A$ or $x \in B$ }.
>(iii) The **complement of $B$ relative to $A$** is the set $A\B :=$ {$x$ : $x \in A$ or $x \notin B$ }.

[Latex Symb 1](https://www.math.utk.edu/~finotti/files/latex/307symb.pdf)

[Latex Symb 2](https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols)

[Latex Symb 3](https://www.caam.rice.edu/~heinken/latex/symbols.pdf)

[Introduction to Real Analysis - Robert G. Bartle | Donald R. Sherbert](https://sciencemathematicseducation.files.wordpress.com/2014/01/0471433314realanalysis4.pdf)

[The Elements of Real Analysis - Robert G. Bartle ](https://hayanihamudi.files.wordpress.com/2014/01/the-elements-of-real-analysis-by-robert-g-bartle.pdf)

1.2 Mathematical Induction
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1.3 Finite and Infinite
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Chapter 2 The Real Numbers
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2.1
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