\part{Real numbers}

Real numbers are special because they're what we called \emph{complete}, a convergent sequence of real numbers always converges to a real number.
Take the sequence of rational numbers $(3,3.1,3.14,3.141,3.1415,...)$ converging to $\pi$, so this is a sequence of numbers in $\mathbb{Q}$ that converges to something not in $\mathbb{Q}$, but in $\mathbb{R}$. Indeed, any convergent real sequence converges to a real number.

Mathematical analysis refers to the study of functions, particularly on domains. Real numbers mix well with analysis; real functions walk the fine line of having remarkable properties while being familiar enough objects to begginers of analysis. Hence we will start our journey through analyis alongside our old friend $\mathbb{R}$.



\chapter{Field of real numbers}




\chapter{Total order of real numbers}
\chapter{Linear combination}


What this book should do
- Introduce concepts of analysis in its generalized form
- Although definitions are defined in generalized form, offer its real analysis correspondence
- Investigate the consequences of real analysis definitions 
# Real analysis
---
1. real numbers
- Field of real numbers
- Total order of real numbers
- Preservation of order under addition
- Preservation of order under multiplication
- Order is complete (Least upper bound property)
- Archimedian property
- R is a 1D Euclidean space
- linear-algebra: triangle inequality
- bounded set
- upper bound
- lower bound
- supremum
- infimum
- absolue value function (metric)
