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# Preliminaries: Numbers, Sets, Proofs, and Bounds

DOI link for Preliminaries: Numbers, Sets, Proofs, and Bounds

Preliminaries: Numbers, Sets, Proofs, and Bounds book

# Preliminaries: Numbers, Sets, Proofs, and Bounds

DOI link for Preliminaries: Numbers, Sets, Proofs, and Bounds

Preliminaries: Numbers, Sets, Proofs, and Bounds book

## ABSTRACT

Real analysis is all about real‐valued functions of a real variable, so an obvious early step is to understand basic properties of the set of real numbers. Many of these properties are related to or inherited from slightly simpler sets, such as the integers and the rational numbers. We reserve some special symbols for these important sets:
^{1}
N
=
\{ 1, 2, 3,
.
.
. \} : the
n
a
t
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r
a
l
n
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b
e
r
s
;
Z
=
\{
.
.
. , - 2, - 1, 0, 1, 2,
.
.
.\} : the integers;
Q
: the rational numbers;
R
⊖
:the
r
e
a
l
n
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.
$$ \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathbb{N}} = {\text{\{ 1, 2, 3, }}{\text{. }}{\text{. }}{\text{. \} : the }}natural{\text{ }}numbers{\text{;}} \hfill \\ {\mathbb{Z}} = {\text{\{ }}{\text{. }}{\text{. }}{\text{. , - 2, - 1, 0, 1, 2, }}{\text{. }}{\text{. }}{\text{.\} : the integers;}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathbb{Q}}{\text{: the rational numbers;}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mathbb{R}\ominus }{\text{:the}}\,real\,numbers. \hfill \\ \end{gathered} $$
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