# Real Analysis/Manual of Style

## Introduction

Welcome to the Real Analysis Wikibook, where we provide meaning behind all of the earlier mathematics that came before and prepare you with the eyes and sense to ready yourself for higher mathematics.

This page will discuss the feel of this book. We will discuss the design layout, how proofs and concepts will be described, and mathematical conventions maintained in the book. This page will have a dual focus of both informing you, the curious burgeoning mathematician, as well as you, the Wikibookians here wondering how to make your mark on this book. As such, this page will have a light-hearted fusion of both editor comments and general reader comments.

To make it easier to skim, we created three headings outlining how this wikibook ought to be designed.

1. Didactic—how the content is told; what the content is
2. Design—how the content looks on the medium (Wikibooks)
3. Description—how the content is presented; how it reads

## Didactic

This wikibook's main headers correspond to the various concepts of elementary mathematics, the name we refer to when discussing the mathematics usually taught at high school or below. Since education mandates that elementary mathematics must be known, this wikibook uses that as a starting ground. However, you may quickly find that although it covers all concepts discussed in elementary mathematics, this wikibook also introduces many new disciplines of mathematics that one may pursue further in their higher mathematical journey. This is good! If you find a certain topic may not be explained with enough rigor, or was brushed off with an axiom, then maybe you can spend time taking a crack at it! However enjoyable it is to prove what were axioms or unexplained phenomenon, remember that some reductions back to axiomatic, informal, or "intuitive" explanations for various theorems or proofs exist for a purpose. These, however sad, are actually meant to shed light on the primary question mathematicians struggle with, and that is what mathematicians can, cannot, or are unable to work without, assuming. This is what this wikibook is meant to teach. Not only is the depth involved with every elementary mathematical concept well trodden, but how much deeper there is left to know in math. Not bad for essentially a group of abstract rules we agreed to follow, huh?

### Audience

Hello! This book is aimed towards the curious mathematicians starting out their journey of mathematics. Preferably, this mathematician should know high school mathematics. This will make most of the concepts at least familiar. However, this wikibook will look at it from a fresh new angle. Since it also contains complex topics buried inside, it is also a refreshing read for advanced mathematicians who want to clear up exactly how concepts relate to one another.

### Interpretation

How should this wikibook be read?

To begin, there are some conventions in terms of chapters:

1. For now, this book is best read sequentially. Concepts build on top of each other the deeper you go.

Have you noticed Appendices or Topics you never heard of before?

1. Some chapters will have an appendix heading, typically located at the bottom. These pages do not have any direct relationship to our assumed audience, since most of the topics will discuss things never-heard-of before. However, they do provide a nice primer to higher mathematics.
2. In fact, some sections will be devoted to higher mathematical concepts. The first chapter will usually be a primer on what the topic is, to the uninitiated. However, the rest of the chapter assumes a level of understanding similar to your understanding of math before you read Analysis, and remember how we assumed you completed secondary education on mathematics—with some knowledge of calculus already—or understood the Calculus wikibook beforehand. It won't look easy!

### Exercises

There are problem sets in this wikibook to complement your learning!

## Design

Although this wikibook is called Real Analysis, it also introduces many other fields of mathematics, ranging from metric spaces, algebra, and set theory as well. This is the primary design focus of this wikibook, in accordance with its didactic. This book ought to introduce, to any budding mathematician, mathematics grander than simply elementary mathematics over real numbers, which quite frankly, is not defined, bounded, or really used in either the exercises or the sections until the very end of the wikibook.

### Templates

This book employs a fairly uniform template style, as well as new styles that should be implemented. Each template will be given its own row to explain itself.

A Common Template usage Rationale
```{{header | title = [[../]] | author = | section = [[../Chapter_Directory|Chapter Title]] | previous = [[../Chapter_Directory_Pre|Chapter Title Pre]] | next = [[../Chapter_Directory_Nxt|Chapter Title Nxt]] | notes = }}``` Headers should be on every chapter. Previous and Next sections should exists only for what is reasonably the next and previous chapter of some overall topic.
`{{TOC|limit=3|float=right}}` Table of contents go on the right, because important links should exists on the right side like a well-designed Wikipedia page. Also the limit is set at 3 so that you can use one large heading and a subheading before time to use the proof heading, which should be hidden.
`{{Math|size=1.2em|Content}}` Should be used for mathematical statements except when complex. They should especially be used for theorem summaries because they can be processed using a search engine.
`{{Equation box 1}}` Should be used to summarize definitions and theorems, with the title in bold and either stating "Theorem" or "Definition of X".