\contentsline {part}{I\hspace {1em}Fundamentals}{1}{part.1}%
\contentsline {chapter}{\numberline {1}Real numbers}{5}{chapter.1}%
\contentsline {section}{\numberline {1.1}Algebraic and order properties of $\mathbb {R}$}{6}{section.1.1}%
\contentsline {section}{\numberline {1.2}Euclidean metric of $\mathbb {R}$}{7}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Open intervals}{7}{subsection.1.2.1}%
\contentsline {section}{\numberline {1.3}Notable properties of $\mathbb {R}$}{8}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Density of $\mathbb {Q}$ in $\mathbb {R}$}{8}{subsection.1.3.1}%
\contentsline {subsection}{\numberline {1.3.2}Archimedean property of $\mathbb {R}$}{9}{subsection.1.3.2}%
\contentsline {chapter}{\numberline {2}Real Sequences and series}{11}{chapter.2}%
\contentsline {section}{\numberline {2.1}Real Sequences}{11}{section.2.1}%
\contentsline {subsection}{\numberline {2.1.1}Bolzano-Weierstrass theorem}{13}{subsection.2.1.1}%
\contentsline {section}{\numberline {2.2}Real Cauchy sequences}{14}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Rationals are not Cauchy complete}{15}{subsection.2.2.1}%
\contentsline {section}{\numberline {2.3}Squeeze theorem}{15}{section.2.3}%
\contentsline {section}{\numberline {2.4}Limit supremum and infimum}{16}{section.2.4}%
\contentsline {section}{\numberline {2.5}Series}{16}{section.2.5}%
\contentsline {section}{\numberline {2.6}Harmonic series}{17}{section.2.6}%
\contentsline {section}{\numberline {2.7}Geometric series}{17}{section.2.7}%
\contentsline {section}{\numberline {2.8}Convergent series tests}{18}{section.2.8}%
\contentsline {chapter}{\numberline {3}Bachmann-Landau family of notations}{21}{chapter.3}%
\contentsline {section}{\numberline {3.1}Original Bachmann-Landau notation}{21}{section.3.1}%
\contentsline {section}{\numberline {3.2}Knuth's notation}{22}{section.3.2}%
\contentsline {section}{\numberline {3.3}Vinogradov-Hardy-Littlewood notation}{23}{section.3.3}%
\contentsline {chapter}{\numberline {4}Continuous function}{25}{chapter.4}%
\contentsline {section}{\numberline {4.1}Limits}{25}{section.4.1}%
\contentsline {section}{\numberline {4.2}Continuous function}{26}{section.4.2}%
\contentsline {section}{\numberline {4.3}Main results}{27}{section.4.3}%
\contentsline {section}{\numberline {4.4}Stronger variants of continuity}{28}{section.4.4}%