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### Basic Natbib Setup and Citation

Source: https://www.overleaf.com/learn/latex/Natbib_citation_styles

This example demonstrates the basic setup of the natbib package, including setting the bibliography style and citation style. It shows how to use \cite and \citep commands with optional arguments for pre- and post-notes.

```latex
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

%Import the natbib package and sets a bibliography  and citation styles
\usepackage{natbib}
\bibliographystyle{abbrvnat}
\setcitestyle{authoryear,open={((},close={))}} %Citation-related commands

\title{Natbib Example}

\author{Overleaf team}

\begin{document}

\maketitle

\section{First Section}
This document is an example with two cited items: \textit{The \LaTeX\ Companion} book \cite[see][chap 2]{latexcompanion} and Einstein's journal paper \citep{einstein}. 


%Imports the bibliography file "sample.bib"
\bibliography{sample}

\end{document}


```

--------------------------------

### Minimal biblatex Setup

Source: https://www.overleaf.com/learn/latex/Basic_bibliography_management

A minimal working example demonstrating the basic setup for the biblatex package. It includes importing the package, adding a bibliography resource, and printing the bibliography.

```latex
\documentclass[letterpaper,10pt]{article}
\usepackage{biblatex} %Imports biblatex package
\addbibresource{sample.bib} %Import the bibliography file

\begin{document}
Let's cite! Einstein's journal paper \cite{einstein} and Dirac's
book \cite{dirac} are physics-related items. 

\printbibliography %Prints bibliography

\end{document}

```

--------------------------------

### Basic natbib setup and citation

Source: https://www.overleaf.com/learn/latex/Bibliography_management_with_natbib

Imports the natbib package and sets the bibliography style to unsrtnat. Includes example citations and bibliography import.

```latex
\usepackage{natbib}
\bibliographystyle{unsrtnat}
\title{Bibliography management: \texttt{natbib} package}
\author{Overleaf}
\date {April 2021}

\begin{document}

\maketitle

This document is an example of \texttt{natbib} package using in bibliography
management. Three items are cited: \textit{The \LaTeX\ Companion} book 
\cite{latexcompanion}, the Einstein journal paper \cite{einstein}, and the 
Donald Knuth's website \cite{knuthwebsite}. The \LaTeX\ related items are 
\cite{latexcompanion,knuthwebsite}. 

\medskip

\bibliography{sample}

\end{document}
```

--------------------------------

### Basic Portuguese Setup with pdfLaTeX

Source: https://www.overleaf.com/learn/latex/Portuguese

This example demonstrates the essential packages and configurations for typesetting Portuguese using pdfLaTeX. Ensure `fontenc` and `babel` are included for correct character rendering and language features. The `hyphenat` package with custom hyphenation rules is also shown.

```latex
\documentclass{article}
% \usepackage[utf8]{inputenc} is no longer required (since 2018)

%Set the font (output) encoding
%--------------------------------------
\usepackage[T1]{fontenc} %Not needed by LuaLaTeX or XeLaTeX
%--------------------------------------

%Portuguese-specific commands
%--------------------------------------
\usepackage[portuguese]{babel}
%--------------------------------------

%Hyphenation rules
%--------------------------------------
\usepackage{hyphenat}
\hyphenation{mate-mática recu-perar}
%--------------------------------------

\begin{document}
\tableofcontents

\vspace{2cm} %Add a 2cm space

\begin{abstract}
Este é um breve resumo do conteúdo do documento escrito em Português.
\end{abstract}

\section{Seção introdutória}
Esta é a primeira seção, podemos acrescentar alguns elementos adicionais 
e tudo será escrito corretamente. Além disso, se uma palavra é um caminho 
muito longo e tem de ser truncado, babel irá tentar truncar corretamente, 
dependendo do idioma.

\section{Segunda seção}
Esta seção é para ver o que acontece com comandos de texto que definem

\[ \lim x =  \theta + 152383.52 \]
\end{document}
```

--------------------------------

### Basic LaTeX Index Creation

Source: https://www.overleaf.com/learn/latex/Indices

This example demonstrates the fundamental setup for creating an index in a LaTeX document. It includes necessary package loading, index generation command, and printing the index.

```latex
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{imakeidx}
\makeindex

\begin{document}
\section{Introduction}
In this example, several keywords\index{keywords} will be used 
which are important and deserve to appear in the Index\index{Index}.

Terms like generate\index{generate} and some\index{others} will 
also show up. 

\printindex
\end{document}

```

```latex
\usepackage{imakeidx}

```

--------------------------------

### Minimal xskak Example: Empty and Initial Chessboards

Source: https://www.overleaf.com/learn/latex/Chess_notation

This example demonstrates the basic usage of the `xskak` package to typeset an empty chessboard and a chessboard with pieces in their initial positions. Ensure `xskak` is loaded before using `\chessboard` and `\newchessgame` commands.

```latex
\documentclass{article}
\usepackage{xskak}
\begin{document}
\chessboard[showmover=false]
\newchessgame
\chessboard
The small white square to the right of the second board is called the \textit{mover}.
\end{document}
```

--------------------------------

### LaTeX Example with Itemized List

Source: https://www.overleaf.com/learn/latex/Commands%23Defining_a_new_command

Illustrates how to create a bulleted list in LaTeX using the itemize environment and the \item command, with an example of customizing list markers.

```latex
\documentclass{article}
\begin{document}
A list example:
\begin{itemize}
  \item[\S] First item
  \item Second item
\end{itemize}
\end{document}


```

--------------------------------

### Demonstrate \marginpar with different alignments

Source: https://www.overleaf.com/learn/latex/Margin_notes

This example showcases the \marginpar command with conditional text for left and right pages, and demonstrates \raggedright and \raggedleft for alignment. It requires the geometry and hyperref packages for page setup and URL support.

```latex
\documentclass[twoside]{article}
\usepackage[a4paper, marginparwidth=75pt, total={10cm, 10cm}]{geometry}
\usepackage{hyperref}
\usepackage{marginnote}
\begin{document}
\section{Lorem Ipsum}
\footnote{Source text: Wikipedia (\url{https://en.wikipedia.org/wiki/Lorem_ipsum})}But I must explain to you how all this mistaken idea of reprobating pleasure and extolling pain arose. To do so, I will give you a complete account of the system, and expound the actual teachings of the great explorer of the truth, the master-builder of human happiness. \marginpar[Note 1: text for left-hand side text]{Note 1: text for right-hand side of pages, it is set justified.} No one rejects, dislikes or avoids pleasure itself, because it is pleasure, but because those who do not know how to pursue pleasure rationally encounter consequences that are extremely painful. Nor again is there anyone who loves or pursues or desires to obtain pain of itself, because it is pain, but occasionally circumstances occur in which toil and pain can procure him some great pleasure.  \marginpar[Note 2: text for left-hand side text]{\raggedright Note 2: text for right-hand side of pages, it is not justified, but uses \texttt{\string\raggedright}.} To take a trivial example, which of us ever undertakes laborious physical exercise, except to obtain some advantage from it? But who has any right to find fault with a man who chooses to enjoy a pleasure that has no annoying consequences, or one who avoids a pain that produces no resultant pleasure? [33] On the other hand, we denounce with righteous indignation and dislike men who are so beguiled and demoralized by the charms of pleasure of the moment, so blinded by desire, that they cannot foresee the pain and trouble that are bound to ensue; and equal blame belongs to those who fail in their duty through weakness of will, which is the same as saying through shrinking from toil and pain. These cases are perfectly simple and easy to distinguish. In a free hour, when our power of choice is untrammeled and when nothing prevents our being able to do what we like best, every pleasure is to be welcomed and every pain avoided. \marginpar[\raggedleft Note 3: text for left-hand side of pages, it is not justified, but uses \texttt{\string\raggedleft}]{Note 3: text for left-hand side of pages}But in certain circumstances and owing to the claims of duty or the obligations of business it will frequently occur that pleasures have to be repudiated and annoyances accepted.  The wise man therefore always holds in these matters to this principle of selection: he rejects pleasures to secure other greater pleasures, or else he endures pains to avoid worse pains.
\end{document}
```

--------------------------------

### Define macros for expansion demonstration

Source: https://www.overleaf.com/learn/latex/Articles/How_does_%5Cexpandafter_work%3A_A_detailed_study_of_consecutive_%5Cexpandafter_commands

Setup macros to test the behavior of \expandafter sequences.

```tex
\def\foo#1{\textbf{#1}}
\def\abc{Hello}
\def\xyz{, World!}
\def\bar{\abc\xyz}
```

--------------------------------

### Basic Nomenclature Example

Source: https://www.overleaf.com/learn/latex/Nomenclatures

This is a fundamental example demonstrating how to use the `nomencl` package to create a simple list of symbols and their descriptions.

```latex
\documentclass{article}
\usepackage{nomencl}
\makenomenclature

\begin{document}
Here is an example:
\nomenclature{\(c\)}{Speed of light in a vacuum}
\nomenclature{\(h\)}{Planck constant}

\printnomenclature
\end{document}
```

--------------------------------

### Example: todonotes Package Usage

Source: https://www.overleaf.com/learn/latex/Questions/Can_I_add_inline_or_margin_comments_to_the_pdf%3F

A comprehensive example demonstrating various todonotes features, including different note styles, colors, font sizes, and inline comments. It also shows how to exclude notes from the list.

```latex
\documentclass[a4paper]{article}
\usepackage[colorlinks]{hyperref}
\usepackage[colorinlistoftodos]{todonotes}
\begin{document}
\listoftodos[A list of things I need to finish]

Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Nulla
\todo{Plain todonotes.} urna. Maecenas interdum nunc in augue. 
Mauris quis massa in ante tincidunt mollis. Proin imperdiet. 
Donec porttitor pede id est. Sed in ante. Integer id arcu. Nam 
lectus nisl, posuere sit amet, imperdiet ut, tristique ac, lorem. 
In erat. In commodo enim. \todo[color=blue!40]{Todonote with a different color.}%
Phasellus libero ipsum, tempor a, pharetra consequat, pellentesque
sit amet, sem. Praesent ut augue luctus elit adipiscing ultricies.
Vestibulum suscipit cursus leo. Nullam molestie justo.

Morbi dui. Morbi convallis mi sed sem. Nulla convallis lacus vitae
risus. Phasellus adipiscing. Nullam tortor. Sed laoreet aliquam
ante. Vestibulum diam. Pellentesque nec leo. Pellentesque velit.
\todo[nolist]{Todonote that is only shown in the margin and not in
the list of todos.}%
Praesent congue mi eu ipsum cursus fringilla. Etiam leo erat,
tristique et, pharetra eget, mollis vitae, velit. In hac habitasse
\todo[size=\small, color=green!40]{A note with a small fontsize.}%
platea dictumst. In quam nibh, facilisis et, laoreet non, facilisis
tempus, justo. Class aptent taciti sociosqu ad litora torquent per
conubia nostra, per inceptos himenaeos.

\todo[inline]{testing testing}

Donec nulla lectus, faucibus sit amet, auctor non, consectetuer
quis, pede. Nullam dictum. Nullam suscipit, ligula in scelerisque
\todo[noline]{A note with no line back to the text.}%
posuere, sapien purus rutrum magna, vitae pharetra leo quam vel
tortor. Donec eleifend condimentum sapien. Etiam sed orci. Aliquam
\todo[inline, color=red!50]{Inline todonotes.}%
tempor. Pellentesque egestas tortor id eros. Donec mauris justo,
commodo id, pellentesque id, eleifend non, mi. Duis venenatis
sagittis metus. Donec tempus metus id lacus. Praesent vel diam.
Morbi nec ante. Vestibulum varius felis ac lacus. Nulla vitae neque
\todo[inline, color=green!40]{A note with no line back to the text.}%
in nibh bibendum volutpat. Quisque accumsan diam. Aenean ultricies
nisl ac lacus. Aliquam posuere. Aenean venenatis tortor in felis.
\end{document}
```

--------------------------------

### Full Set of Text Formatting Examples in LaTeX

Source: https://www.overleaf.com/learn/latex/Bold%2C_italics_and_underlining

A collection of various text formatting examples including bold, italics, underline, and emphasis, demonstrating their usage and nesting.

```latex
First example, bold, italics and underline:

Some of the \\textbf{greatest} discoveries in \\underline{science} were made by \\textbf{\\\\emph{accidentப்பட்டன}}.

\\vspace{1.5cm}

Example of italicized text: 

Some of the greatest discoveries in science were made by \\emph{accident}.

\\vspace{1.5cm}

Example of boldface text:

Some of the \\textbf{greatest} discoveries in science were made by accident.

\\vspace{1.5cm}

Example of underlined text:

Some of the greatest discoveries in \\underline{science} were made by accident.

\\vspace{1.5cm}

Example of emphasized text in different contexts:

Some of the greatest \\emph{discoveries} in science were made by accident.

\\textit{Some of the greatest \\emph{discoveries} in science were made by accident.}

\\textbf{Some of the greatest \\emph{discoveries} in science were made by accident.}
```

--------------------------------

### LaTeX: Full Document Example

Source: https://www.overleaf.com/learn/latex/Font_sizes%2C_families%2C_and_styles

A comprehensive example combining various font size, family, and style commands within a complete LaTeX document structure.

```latex
\documentclass{article}
\begin{document}
%Example of different font sizes and types
This is a simple example, {\tiny this will show different font sizes} and also \textsc{different font styles}.

\vspace{1cm}

%Example of different font sizes and types
In this example the {\huge huge font size} is set and the {\footnotesize Foot note size also}. There's a fairly large set of font sizes.

\vspace{1cm}

%Example of different font sizes and types
In this example, a command and a switch are used. \texttt{A command is used to change the style of a sentence}.

\sffamily
A switch changes the style from this point to the end of the document unless another switch is used.
\rmfamily

\vspace{1cm}

%Example of different font sizes and types
Part of this text is written \textsl{in different font style} to highlight it.
\end{document}

```

--------------------------------

### Minimal biblatex Example

Source: https://www.overleaf.com/learn/latex/Bibliography_management_with_biblatex

A basic LaTeX document structure demonstrating the minimal setup for using the biblatex package to cite and print a bibliography.

```latex
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage{biblatex}
\addbibresource{sample.bib}

\begin{document}
Let's cite! The Einstein's journal paper \cite{einstein} and the Dirac's 
book \cite{dirac} are physics related items. 

\printbibliography

\end{document>

```

--------------------------------

### Example of 'blank' pages in a book class document

Source: https://www.overleaf.com/learn/latex/Headers_and_footers

This example demonstrates how LaTeX inserts a 'blank' page (page 2) to ensure the next chapter starts on a right-hand page when using the `book` class without `openany`. This page may still contain headers.

```latex
\begin{document}

\frontmatter
\maketitle
This is frontmatter which uses Roman numerals.

\mainmatter
\chapter{Where do I start}
Chapter 1: A short chapter that ends on page 1.

\chapter{Things I remember}
Chapter 2: Starts on page 3, so \LaTeX{} inserts a ``blank'' page 2.

\end{document}


```

--------------------------------

### Calling the Example TeX Macro

Source: https://www.overleaf.com/learn/latex/Articles/What_is_a_TeX_token_list

Demonstrates how to invoke the previously defined `\mymacro` with appropriate arguments to produce output.

```tex
\mymacro abc THIS TEXT defz

```

--------------------------------

### Example pTeX Document with latexmkrc

Source: https://www.overleaf.com/learn/latex/Latex-questions/Does_Overleaf_support_pTeX%3F

This snippet shows a complete Overleaf project setup including the main LaTeX file and the `latexmkrc` configuration for pTeX.

```latex
\documentclass{jsarticle}

\bibliographystyle{jplain}
\title{p\LaTeX\ 実験}
\author{林蓮枝}

\begin{document}

\maketitle

\begin{abstract}
本稿では、文書組版システムp\LaTeX{}の使い方を解説します。p\LaTeX{}を利用するときには、あらかじめ文章中に\TeX{}コマンドと呼ばれる組版用の指示を混在させ\ldots
\end{abstract}

\section{導入}
こんにちは世界！

\end{document}
```

```bash
$latex = 'platex';
$bibtex = 'pbibtex';
$dvipdf = 'dvipdfmx %O -o %D %S';
```

--------------------------------

### Incorrect Display Math Termination with Single $

Source: https://www.overleaf.com/learn/latex/Errors/Display_math_should_end_with_%24%24

This example shows how starting display math with '$$' and ending with a single '$' causes the 'Display math should end with $$' error.

```latex
\documentclass{article}
\usepackage[textwidth=8cm]{geometry}
\begin{document}
\noindent \verb|$$ E=mc^2$| generates an error because the math is 
started by \texttt{\$\\$} but terminated by a single \texttt{\$}:

$$ E=mc^2$

\noindent\verb|$$ E=mc^2$ $| also generates an error because of the space between
the terminating \texttt{\$} characters:

$$ E=mc^2$ $
\end{document}

```

--------------------------------

### Example Usage of Numbered Environments

Source: https://www.overleaf.com/learn/latex/Environments%23Defining_a_new_environment

Demonstrates using both manually defined and \newtheorem-based numbered environments, showing how numbering resets with sections.

```latex
\documentclass{article}
% Define our numbered environment within the preamble
\newcounter{example}[section]
\newenvironment{example}[1][]
{\refstepcounter{example}\par\medskip
   \noindent \textbf{Example~\theexample. #1} \rmfamily}{\medskip}

% Another numbered environment defined with \newtheorem
\usepackage{amsmath} % For the \newtheorem command
\newtheorem{SampleEnv}{Sample Environment}[section]
\begin{document}

\section{User-defined numbered environments}

\begin{example}
First user-defined numbered environment (number \theexample).
\end{example}

\begin{example}
Second user-defined numbered environment (number \theexample).
\end{example}

\section{More user-defined numbered environments}
Note how the example numbering has restarted at 1:

\begin{example}
First user-defined numbered environment (number \theexample).
\end{example}

\begin{SampleEnv}
User-defined environment created with the \verb|\newtheorem| command.
\end{SampleEnv}
\end{document}
```

--------------------------------

### Enable SyncTeX with compressed output

Source: https://www.overleaf.com/learn/latex/MLTeX_SyncTeX_and_EncTeX_TeX_extensions

Use the -synctex=1 option to enable SyncTeX and generate a compressed .synctex.gz output file.

```bash
pdftex -synctex=1 myfile.tex

```

--------------------------------

### Minimal Beamer Presentation Example

Source: https://www.overleaf.com/learn/latex/Beamer

A basic Beamer document structure to create a title page and a content frame. Use this as a starting point for simple presentations.

```latex
\documentclass{beamer}
%Information to be included in the title page:
\title{Sample title}
\author{Anonymous}
\institute{Overleaf}
\date{2021}

\begin{document}

\frame{\titlepage}

\begin{frame}
\frametitle{Sample frame title}
This is some text in the first frame. This is some text in the first frame. This is some text in the first frame.
\end{frame}

\end{document}

```

--------------------------------

### Example Usage of Boxed Environment

Source: https://www.overleaf.com/learn/latex/Environments%23Defining_a_new_environment

Demonstrates using the 'boxed' environment with and without providing a value for the optional first argument.

```latex
\documentclass{article}
% Note the default value for the first
% argument is provided by [This is a box]
\newenvironment{boxed}[2][This is a box]{\begin{center}
    Argument 1 (\#1)=#1\[1ex]
    \begin{tabular}{|p{0.9\textwidth}|}
    \hline\
    Argument 2 (\#2)=#2\[2ex]
    }
    { 
    \\\\hline
    \end{tabular} 
    \end{center}
    }
\begin{document}
\textbf{Example 1}: Use the default value for the first argument:
 
\begin{boxed}{Some preliminary text}
This text is \textit{inside} the environment.
\end{boxed}

This text is \textit{outside} the environment.

\vskip12pt

\textbf{Example 2}: Provide a value for the first argument:
 
\begin{boxed}[This is not the default value]{Some more preliminary text}
This text is still \textit{inside} the environment.
\end{boxed}

This text is also \textit{outside} the environment.
\end{document}
```

--------------------------------

### Demonstrating escape sequence errors

Source: https://www.overleaf.com/learn/latex/Articles/An_Introduction_to_LuaTeX_%28Part_2%29%3A_Understanding_%5Cdirectlua

An example showing how standard double quotes cause errors when a TeX command starts with a character that forms a valid Lua escape sequence (like \n).

```latex
\directlua{
   tex.print("\newtest{Hello}")
}
```

--------------------------------

### Full LaTeX document demonstrating math styles

Source: https://www.overleaf.com/learn/latex/Display_style_in_math_mode

A complete document example showing inline and display math alongside various style overrides.

```latex
\documentclass{article}
\usepackage{amsmath}
\title{Exploring math display styles}
\author{Overleaf team}
\begin{document}
\maketitle
Depending on the value of \(x\) the equation \( f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \) may diverge or converge.

\[ f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \]

\vspace{1cm}

Inline maths elements can be set with a different style: \(f(x) = \displaystyle \frac{1}{1+x}\). The same is true for display math material:

\begin{align*}
f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \\
\textstyle f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \\
\scriptstyle f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x} \\
\scriptscriptstyle f(x) = \sum_{i=0}^{n} \frac{a_i}{1+x}
\end{align*}
\end{document}
```

--------------------------------

### Incorrect \verb command with mismatched delimiters

Source: https://www.overleaf.com/learn/latex/Errors/LaTeX_Error%3A_%5Cverb_ended_by_end_of_line

This example shows incorrect usage of the \verb command where the opening delimiter is omitted or mismatched, leading to errors. Always ensure the start and end delimiters are identical and present.

```latex
\verb{\frac{1}{2}}
```

```latex
\verb \frac{1}{2}
```

```latex
{\verb \frac{1}{2}}
```

```latex
\verb!\frac{1}{2}
```

--------------------------------

### Correct \verb command usage with delimiters

Source: https://www.overleaf.com/learn/latex/Errors/LaTeX_Error%3A_%5Cverb_ended_by_end_of_line

The \verb command requires matching start and end delimiters. Any character can be used as a delimiter, provided it does not appear within the text to be typeset verbatim. This example uses '!' as the delimiter.

```latex
\verb!\frac{1}{2}!
```

--------------------------------

### Mix Arabic (RTL) and LTR languages with LuaLaTeX

Source: https://www.overleaf.com/learn/latex/Multilingual_typesetting_on_Overleaf_using_babel_and_fontspec

Use babel with the bidi=basic option for LuaLaTeX, which offers comprehensive support for complex scripts and bidirectional typesetting. This example includes Arabic text and basic font setup.

```latex
% Arabic text in this example is from 
% https://github.com/latex3/babel/blob/main/samples/lua-arabic.tex
\documentclass{article}
\usepackage[english,bidi=basic]{babel}
\babelprovide[import,main]{arabic}
\babelfont{rm}{FreeSerif}
\babelfont{sf}{FreeSans}
\babelfont{tt}{FreeMono}
\begin{document}
الكهرمان اسمه باليونانية الإيلقطرون[3] (معرب ἤλεκτρον إيلكترون أي ذو
البريق، ومنه الإلكترون عند الفيزيائيين، وعليه تسمية الكهرباء في
الفارسية برق)، واشتق منه اسم فاعليتيه فسمي إلكترسمس (ηλεκτρ‌ισμός)
للدلالة على الكهرباء. أما باللاتينية فالكلمة للكهرباء هي إيلكترستاس
(ēlectricitās)، وهي مشتقة من إيلكتركس (ēlectricus) أي شبيه الكهرمان.
\end{document}

```

--------------------------------

### Basic Multicolumn Layout

Source: https://www.overleaf.com/learn/latex/Multiple_columns%23Inserting_vertical_rulers

This example shows how to set up a basic three-column layout using the `multicols` environment. The optional header text can include sections and paragraphs.

```latex
\documentclass{article}
\usepackage{blindtext}
\usepackage{multicol}
\title{Multicols Demo}
\author{Overleaf}
\date{April 2021}

\begin{document}
\maketitle

\begin{multicols}{3}
[
\section{First Section}
All human things are subject to decay. And when fate summons, Monarchs must obey.
]
\blindtext\blindtext
\end{multicols}

\end{document}
```

--------------------------------

### Create Proof Environments with amsthm

Source: https://www.overleaf.com/learn/latex/Theorems_and_proofs

Utilize the 'proof' environment provided by the amsthm package to visually distinguish mathematical proofs from regular text. This example also shows defining a 'lemma' environment.

```latex
\documentclass{article}
\usepackage[english]{babel}
\usepackage{amsthm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\begin{document}
\section{Introduction}
\begin{lemma}
Given two line segments whose lengths are \(a\) and \(b\) respectively there 
is a real number \(r\) such that \(b=ra\).
\end{lemma}

\begin{proof}
To prove it by contradiction try and assume that the statement is false,
proceed from there and at some point you will arrive to a contradiction.
\end{proof}
\end{document}
```

--------------------------------

### Use a Custom LaTeX Package

Source: https://www.overleaf.com/learn/latex/Writing_your_own_package

This document demonstrates how to use the custom 'examplepackage' with the 'red' option. It includes the necessary \documentclass, \usepackage, and \makeindex commands, and shows how to use the custom 'example' environment and '\important' command.

```tex
\documentclass{article}
\usepackage[utf8]{inputenc}

\usepackage[red]{examplepackage}

\makeindex

\title{Package Example}
\author{Team Learn ShareLaTeX}
\date{ }

\begin{document}

\maketitle

\section{Introduction}
In this document a new package is tested. This package allows special numbered
environments

\begin{example}
This text is inside a special environment, some boldface text is printed
at the beginning and a new indentation is set.
\end{example}

Also, there's a special command for \important{important!words} that will be
printed in a special \important{colour} depending on the parameter used in the
\important{package}
importation statement. Because it's \important{important}.

\printindex

\end{document}
```

--------------------------------

### Include a LaTeX file using \include

Source: https://www.overleaf.com/learn/latex/Management_in_a_large_project%23Inputting_and_including_files

Use \include{filename} to insert the content of filename.tex. The file should not contain LaTeX preamble code. LaTeX will start a new page before processing the inputted material. Each \include'd file gets its own .aux file.

```latex
\include{filename}
```

--------------------------------

### Display hb-view help documentation

Source: https://www.overleaf.com/learn/latex/Articles/An_overview_of_technologies_supporting_the_use_of_colour_emoji_fonts_in_LaTeX

Displays all available command-line options for the hb-view utility.

```bash
hb-view --help-all
```

--------------------------------

### Example TeX Macro Definition

Source: https://www.overleaf.com/learn/latex/Articles/What_is_a_TeX_token_list

A concrete example of defining a TeX macro named `\mymacro` with specific delimiters and a parameter.

```tex
\def\mymacro abc #1 defz{I typed "#1"!}

```

--------------------------------

### Complete LaTeX Document Example

Source: https://www.overleaf.com/learn/latex/Integrals%2C_sums_and_limits

A full LaTeX document demonstrating integrals, multiple integrals, sums, products, and limits, including necessary packages and sections.

```latex
\documentclass{article}
\title{Integrals, Sums and Limits}
\author{Overleaf}
\date{}
\usepackage{amsmath}

\begin{document}

\maketitle

\section{Integrals}

Integral \(\int_{a}^{b} x^2 dx\) inside text.

\medskip

The same integral on display:
\[
    \int_{a}^{b} x^2 \,dx
\]
and multiple integrals:
\begin{gather*}
    \iint_V \mu(u,v) \,du\,dv
\
    \iiint_V \mu(u,v,w) \,du\,dv\,dw
\
    \iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\
    \idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\
    \oint_V f(s) \,ds
\end{gather*}

\section{Sums and products}

Sum \(\sum_{n=1}^{\infty} 2^{-n} = 1\) inside text.

The same sum on display:
\[
    \sum_{n=1}^{\infty} 2^{-n} = 1
\]

Product \(\prod_{i=a}^{b} f(i)\) inside text.

The same product on display:
\[
    \prod_{i=a}^{b} f(i)
\]

\section{Limits}

Limit \(\lim_{x\to\infty} f(x)\) inside text.

The same limit on display:
\[
    \lim_{x\to\infty} f(x)
\]

\end{document}
```

--------------------------------

### Full document example for leftskip and rightskip

Source: https://www.overleaf.com/learn/latex/Paragraph_formatting

A complete LaTeX document demonstrating the effect of flexible glue on paragraph alignment.

```latex
\documentclass{article}
\title{leftskip and rightskip examples}
% Choose a conveniently small page size
\usepackage[paperheight==16cm,paperwidth=12cm,textwidth=8cm]{geometry}
% Set the value of some paragraph-related parameters
\setlength{\lineskip}{3.5pt}
\setlength{\lineskiplimit}{2pt}
\setlength{\parindent}{20pt}
\setlength{\baselineskip}{12pt}

\begin{document}
\input text.tex % A generated TeX file which defines the macros
% \mytextA and \mytextB which define text for two paragraphs
% Typeset a paragraph with default values of \leftskip and \rightskip

\mytextB

% Set \leftskip and \rightskip to "extremely flexible" glue settings
\leftskip 0pt plus 1fill
\rightskip 0pt plus 1fill
\vspace{20pt}

% Now re-typeset the same paragraph
\mytextB

\end{document}
```

--------------------------------

### Directlua Tilde Error Example

Source: https://www.overleaf.com/learn/latex/Articles/An_Introduction_to_LuaTeX_%28Part_2%29%3A_Understanding_%5Cdirectlua

Example of code that fails when passed directly to \directlua because the tilde is treated as an active character by TeX.

```latex
\directlua{
   local x=3
   if x ~= 4 then
   print("x is not equal to 4")
   end
}
```

--------------------------------

### Enable SyncTeX: Output .synctex

Source: https://www.overleaf.com/learn/latex/MLTeX_EncTeX_and_SyncTeX_TeX_extensions

Use this command-line option to enable SyncTeX and generate an uncompressed synchronization file (.synctex).

```bash
pdftex -synctex=-1 myfile.tex
```

--------------------------------

### Markdown to HTML Conversion Example

Source: https://www.overleaf.com/learn/latex/Articles/How_to_write_in_Markdown_on_Overleaf

Shows the resulting HTML output after converting the example Markdown text. This illustrates the transformation process.

```html
<h1 id="grocerylist">Grocery list</h1>
<p><em>Remember</em> to grab as much as we can during upcoming <a href="http://acme-marg.com">sales</a>!</p>
<h2 id="food">Food</h2>
<ul>
<li>baked beans</li>
<li>spaghetti</li>
</ul>
<h2 id="stationery">Stationery</h2>
<ul>
<li>writing pad</li>
<li>pencils</li>
</ul>

```

--------------------------------

### Pre-content check for mark variables

Source: https://www.overleaf.com/learn/latex/Articles/How_does_LaTeX_typeset_headers_and_footers%3F

Initializes \topmark and \firstmark based on the \botmark value from the previous page.

```pseudocode
if(\botmark != NULL)
{
   \topmark = \botmark;
   \firstmark= NULL;
}
```

--------------------------------

### Example with French Quotation Marks using dirtytalk

Source: https://www.overleaf.com/learn/latex/Typesetting_quotations

An example demonstrating the `dirtytalk` package with French quotation marks, including nested quotes.

```latex
\documentclass{article}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}

\usepackage[
    left = \flqq{},
    right = \frqq{},
    leftsub = \flq{},
    rightsub = \frq{}
]{dirtytalk}

\begin{document}
\section{Introduction}

Typing quotations with this package is quite easy:

\say{Here, a quotation is written and even some \say{nested} quotations are possible}
\end{document}
```

--------------------------------

### Full Spanish LaTeX Document Example

Source: https://www.overleaf.com/learn/latex/Spanish

A complete example demonstrating `fontenc` and `babel` for Spanish, including localized mathematical commands and quotation marks.

```latex
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[spanish]{babel}
\begin{document}
\section{Sección con teoremas}
Esta sección es para ver que pasa con los comandos que definen texto

\[ \lim x =  \tg {\theta} + \max \{3.52, 4.22 \} \]

El paquete también agrega un comportamiento especial a <<éstas márcas para hacer citas textuales>> tal como lo indican las relgas de la RAE.
\end{document}
```

--------------------------------

### Enable SyncTeX: Output .synctex.gz

Source: https://www.overleaf.com/learn/latex/MLTeX_EncTeX_and_SyncTeX_TeX_extensions

Use this command-line option to enable SyncTeX and generate a gzipped synchronization file (.synctex.gz).

```bash
pdftex -synctex=1 myfile.tex
```

--------------------------------

### Constructing a Fixed-Width Horizontal Box with Flexible Glue

Source: https://www.overleaf.com/learn/latex/Articles/Pandora%E2%80%99s_%5Chbox%3A_Using_LuaTeX_to_Lift_the_Lid_of_TeX_Boxes

This example demonstrates creating a 100pt wide \hbox. It uses \hskip with flexible 'plus' and 'fill' components to distribute space and ensure the box reaches the target width.

```tex
\hbox to100pt{
A\hskip4pt plus3pt minus 2pt B% 
\hskip 0pt plus 2fil C% 
\hskip 0pt plus 2fill D% 
\hskip 0pt plus 3fill}
```

--------------------------------

### LaTeX Example of Undefined References

Source: https://www.overleaf.com/learn/latex/Errors/There_were_undefined_references

This example demonstrates how typos in \ref{...} or referencing undefined labels lead to 'undefined references' errors in LaTeX.

```latex
\section{introduction}\label{introduction}
A typo when referencing the introduction would be \ref{intorduction}.

Another error is referencing a label which has never been defined such as \ref{section1}

```

--------------------------------

### LaTeX Error Message Example

Source: https://www.overleaf.com/learn/latex/Errors/LaTeX_Error%3A_Missing_%5Cbegindocument

This is an example of the error message generated when \begin{document} is missing. Ensure \begin{document} is included after the preamble to resolve this.

```latex
main.tex, line 5
LaTeX Error: Missing \begin{document}.
See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ... You're in trouble here. Try typing <return> to proceed. If that doesn't work, type X <return> to quit.
```

--------------------------------

### Use Custom LaTeX Environment with Default and Provided Arguments

Source: https://www.overleaf.com/learn/latex/Environments

Demonstrates using the custom 'boxed' environment. The first example uses the default value for the optional argument, while the second provides a custom value.

```latex
\documentclass{article}
% Note the default value for the first
% argument is provided by [This is a box]
\newenvironment{boxed}[2][This is a box]{\begin{center}
    Argument 1 (\#1)=#1\[1ex]
    \begin{tabular}{|p{0.9\textwidth}|}
    \hline\
    Argument 2 (\#2)=#2\[2ex]
    }
    { 
    \\\hline
    \end{tabular} 
    \end{center}
    }
\begin{document}
\textbf{Example 1}: Use the default value for the first argument:
 
\begin{boxed}{Some preliminary text}
This text is \textit{inside} the environment.
\end{boxed}

This text is \textit{outside} the environment.

\vskip12pt

\textbf{Example 2}: Provide a value for the first argument: 
 
\begin{boxed}[This is not the default value]{Some more preliminary text}
This text is still \textit{inside} the environment.
\end{boxed}

This text is also \textit{outside} the environment.
\end{document}
```

--------------------------------

### TeX Error Message Example

Source: https://www.overleaf.com/learn/latex/Articles/What_is_a_TeX_token_list

An example of a typical TeX error message that can occur when macro delimiters do not match due to category code changes.

```tex
Runaway argument?
THIS TEXT defz 
! Paragraph ended before \mymacro was complete.
<to be read again>
\par
l.22

```

--------------------------------

### Basic Hebrew Document Setup

Source: https://www.overleaf.com/learn/latex/Multilingual_typesetting_on_Overleaf_using_polyglossia_and_fontspec

A minimal LaTeX document setup for Hebrew, including setting the default language to Hebrew and defining a Hebrew font.

```latex
\documentclass{article}
\usepackage{polyglossia}
\setdefaultlanguage[numerals=hebrew]{hebrew}
\setotherlanguage{english}
\newfontfamily\hebrewfont[Script=Hebrew]{Hadasim CLM}
\begin{document}
\section{מבוא}
זוהי עובדה מבוססות שדעתו של הקורא תהיה מוסחת עלידי טקטס קריא כאשר הוא יביט בפריסתו.  -
```

--------------------------------

### Correct verbatim text with \texttt{verbatim} environment (fraction example)

Source: https://www.overleaf.com/learn/latex/Errors/LaTeX_Error%3A_%5Cverb_ended_by_end_of_line

This demonstrates using the \texttt{verbatim} environment to correctly display mathematical fractions or other code-like text that might otherwise cause issues with \verb commands.

```latex
\begin{verbatim}
\frac{1}{2}
\end{verbatim}
```

--------------------------------

### Create a Simple Glossary in LaTeX

Source: https://www.overleaf.com/learn/latex/Glossaries

Import the glossaries package and use \newglossaryentry to define terms. Use \gls to reference terms and \printglossaries to display the glossary.

```latex
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{glossaries}

\makeglossaries

\newglossaryentry{latex}
{
    name=latex,
    description={Is a markup language specially suited 
    for scientific documents}
}

\newglossaryentry{maths}
{
    name=mathematics,
    description={Mathematics is what mathematicians do}
}

\title{How to create a glossary}
\author{ }
\date{ }

\begin{document}
\maketitle

The \Gls{latex} typesetting markup language is specially suitable 
for documents that include \gls{maths}. 

\clearpage

\printglossaries

\end{document>
```

--------------------------------

### Create a Description List with description

Source: https://www.overleaf.com/learn/latex/Lists

The `description` environment creates lists where each item can have a custom label specified in square brackets after the `\item` command.

```latex
\documentclass{article}
\usepackage[english]{babel} % To obtain English text with the blindtext package
\usepackage{blindtext}
\begin{document}

\begin{description}
   \item This is an entry \textit{without} a label.
   \item[Something short] A short one-line description.
   \item[Something long] A much longer description. \blindtext[1]
\end{description}
\end{document}

```

--------------------------------

### Complete LaTeX Document with Font Examples

Source: https://www.overleaf.com/learn/latex/Font_sizes_and_kinds%23Reference_guide

A full LaTeX document example combining various font size, family, and style commands demonstrated in the preceding sections.

```latex
\documentclass{article}
\begin{document}
%Example of different font sizes and types
This is a simple example, {\tiny this will show different font sizes} and also \textsc{different font styles}. 

\vspace{1cm}

%Example of different font sizes and types
In this example the {\huge huge font size} is set and the {\footnotesize Foot note size also}. There's a fairly large set of font sizes.

\vspace{1cm}

%Example of different font sizes and types
In this example, a command and a switch are used. \texttt{A command is used to change the style of a sentence}.

\sffamily
A switch changes the style from this point to the end of the document unless another switch is used.
\rmfamily

\vspace{1cm}

%Example of different font sizes and types
Part of this text is written \textsl{in different font style} to highlight it.
\end{document}
```

--------------------------------

### Compare doublespacing and spacing environment

Source: https://www.overleaf.com/learn/latex/Articles/How_to_change_paragraph_spacing_in_LaTeX

This example demonstrates that using the \doublespacing command and the spacing environment with a calculated stretch value results in identical baselineskip values.

```latex
\begingroup
\doublespacing
\verb|\baselineskip|=\texttt{\the\baselineskip} (via \verb|\doublespacing)|

\blindtext[1]

\endgroup

\begin{spacing}{1.618}
\verb|\baselineskip|=\texttt{\the\baselineskip} (via \verb|\begin{spacing}{1.618})|

\blindtext[1]
\end{spacing}
```

--------------------------------

### Example of numbered theorems, corollaries, and lemmas in LaTeX

Source: https://www.overleaf.com/learn/latex/Theorems_and_proofs

Demonstrates defining and using theorem-like environments with custom numbering schemes. Includes theorems numbered by section, corollaries reset by theorems, and lemmas sharing the theorem counter.

```latex
\documentclass{article}
\usepackage[english]{babel}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{lemma}[theorem]{Lemma}

\begin{document}
\section{Introduction}
Theorems can easily be defined:

\begin{theorem}
Let \(f\) be a function whose derivative exists in every point, then \(f\) is 
a continuous function.
\end{theorem}

\begin{theorem}[Pythagorean theorem]
\label{pythagorean}
This is a theorem about right triangles and can be summarised in the next 
equation 
\[ x^2 + y^2 = z^2 \]
\end{theorem}

And a consequence of theorem \ref{pythagorean} is the statement in the next 
corollary.

\begin{corollary}
There's no right rectangle whose sides measure 3cm, 4cm, and 6cm.
\end{corollary}

You can reference theorems such as \ref{pythagorean} when a label is assigned.

\begin{lemma}
Given two line segments whose lengths are \(a\) and \(b\) respectively there is a 
real number \(r\) such that \(b=ra\).
\end{lemma}

```
Tokens:
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