$$ \def\R{\mathbb R} \def\N{\mathbb N} \def\Q{\mathbb Q} \def\Z{\mathbb Z} \newcommand{\st}{~:~} \newcommand{\cee}{\mathsf{C}} \newcommand{\bee}{\mathrm{B}} \def\mathhlyellow#1{\bbox[##fff3bf,2px,border:1px solid ##b7791f]{#1}} \def\mathhlgold#1{\bbox[##ffe8a3,2px,border:1px solid ##b7791f]{#1}} \def\mathhlorange#1{\bbox[##ffddb3,2px,border:1px solid ##c05621]{#1}} \def\mathhlgreen#1{\bbox[##d9f7d7,2px,border:1px solid ##3f8f46]{#1}} \def\mathhlmint#1{\bbox[##d4f5e9,2px,border:1px solid ##2c7a7b]{#1}} \def\mathhlblue#1{\bbox[##dceeff,2px,border:1px solid ##2b6cb0]{#1}} \def\mathhlpurple#1{\bbox[##eadffd,2px,border:1px solid ##6b46c1]{#1}} \def\mathhlpink#1{\bbox[##ffdbe8,2px,border:1px solid ##b83280]{#1}} \def\mathhlred#1{\bbox[##ffd6d6,2px,border:1px solid ##c53030]{#1}} \def\mathhlgray#1{\bbox[##e8ecef,2px,border:1px solid ##64748b]{#1}} \def\mathhlgrey#1{\bbox[##e8ecef,2px,border:1px solid ##64748b]{#1}} $$
References
An Illustrated Course in Real Analysis
Welcome
1
Introduction
The Real Numbers
2
The Field and Order Axioms
3
The Least Upper Bound Axiom
4
Consequences of the Least Upper Bound Axiom
Metric Spaces
5
Metric Spaces and Distance Functions
6
\(\mathbb{R}\)
as a Metric Space
7
Open Sets
8
Properties of Open Sets
9
Boundedness
References
Appendices
A
Proof-Writing Resources
B
Collections of Sets
C
The Cauchy-Schwarz Inequality
D
The Well-Ordering Property
E
Links to Desmos Web Apps
References
Abbott, Stephen. 2016.
Understanding Analysis
. Vol. 2. Springer.
Rock, John A. 2025.
Arbitrarily Close
. 619 Wreath.
Schumacher, Carol. 2008.
Closer and Closer: Introducing Real Analysis
. Jones
and
Bartlett Publishers.
9
Boundedness
A
Proof-Writing Resources