LaTeX lecture notes template

A LaTeX lecture notes template for a course handout, with numbered theorem environments and one section per lecture.

First page of the Lecture notes template

Open in LaTeX.to Download all files (zip)

Engine: pdfLaTeX
Bibliography: none
License: Public domain, use freely.

Files:

main.tex

\documentclass[11pt,a4paper]{article}

\usepackage[margin=2.5cm,headheight=14pt]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{tikz}
\usepackage{fancyhdr}
\usepackage{hyperref}  % keep hyperref last

% All of these share one counter, so the numbering runs 1.1, 1.2, 1.3 through
% a section whatever kind of block comes next.
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}[definition]{Example}
\theoremstyle{plain}
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{lemma}[definition]{Lemma}
\theoremstyle{remark}
\newtheorem{remark}[definition]{Remark}

\pagestyle{fancy}
\fancyhf{}
\fancyhead[L]{Course 101}
\fancyhead[R]{Lecture Notes}
\fancyfoot[C]{\thepage}

% Course, lecturer and term.
\title{Course 101: Introduction to Real Analysis \\ Lecture Notes}
\author{Your Name \\ University of Somewhere}
\date{Fall Term}

\begin{document}
\maketitle
\thispagestyle{fancy}
\tableofcontents
\newpage

\section{Lecture 1: Sequences and Limits}

A sequence is an infinite list of real numbers $a_1, a_2, a_3, \dots$, written
$(a_n)$. The whole of this lecture is about one question: what it means for
such a list to settle down on a value.

\begin{definition}[Convergence]
  A sequence $(a_n)$ converges to $L \in \mathbb{R}$ if for every
  $\varepsilon > 0$ there is an $N \in \mathbb{N}$ such that
  \begin{equation} \label{eq:convergence}
    |a_n - L| < \varepsilon \quad \text{for all } n \ge N .
  \end{equation}
  We write $\lim_{n \to \infty} a_n = L$, or $a_n \to L$.
\end{definition}

Read \eqref{eq:convergence} as a game: your opponent names a tolerance
$\varepsilon$, and you have to name a point $N$ in the sequence beyond which
every term is within that tolerance of $L$. Figure~\ref{fig:neighborhood}
shows the picture on the number line.

\begin{figure}[htbp]
  \centering
  \begin{tikzpicture}
    \draw[->] (-0.5,0) -- (8.5,0) node[right] {$\mathbb{R}$};
    \foreach \x/\lbl in {2/{L - \varepsilon}, 4/{L}, 6/{L + \varepsilon}} {
      \draw (\x,0.15) -- (\x,-0.15) node[below] {$\lbl$};
    }
    \fill (4,0) circle (1.6pt);
    \draw[very thick] (2,0.5) -- (6,0.5);
    \node[above] at (4,0.55) {$a_n$ for every $n \ge N$};
  \end{tikzpicture}
  \caption{Beyond the index $N$, the whole tail of the sequence lies inside
    the band of width $2\varepsilon$ around $L$.}
  \label{fig:neighborhood}
\end{figure}

\begin{example}
  The sequence $a_n = 1/n$ converges to $0$. Given $\varepsilon > 0$, choose
  any $N$ larger than $1/\varepsilon$. Then for $n \ge N$ we have
  $|1/n - 0| = 1/n \le 1/N < \varepsilon$.
\end{example}

\begin{theorem}[Uniqueness of the limit]
  A sequence has at most one limit.
\end{theorem}

\begin{proof}
  Suppose $a_n \to L$ and $a_n \to M$ with $L \neq M$. Take
  $\varepsilon = |L - M| / 2$. Beyond some index every term is within
  $\varepsilon$ of $L$, and beyond some other index every term is within
  $\varepsilon$ of $M$. Past both indices a single term is within
  $\varepsilon$ of each, so $|L - M| < 2\varepsilon = |L - M|$, which is
  impossible.
\end{proof}

\begin{remark}
  The definition never mentions how fast the terms approach $L$. Convergence
  is about what happens in the end, so changing finitely many terms of a
  sequence changes neither whether it converges nor what it converges to.
\end{remark}

\section{Lecture 2: Continuity}

Continuity is the same idea moved from a sequence to a function: small changes
in the input are to make only small changes in the output.

\begin{definition}[Continuity at a point]
  A function $f : \mathbb{R} \to \mathbb{R}$ is continuous at $c$ if for every
  $\varepsilon > 0$ there is a $\delta > 0$ such that $|x - c| < \delta$
  implies $|f(x) - f(c)| < \varepsilon$.
\end{definition}

\begin{lemma}[Sequential criterion]
  $f$ is continuous at $c$ if and only if $f(a_n) \to f(c)$ for every sequence
  $(a_n)$ with $a_n \to c$.
\end{lemma}

This lemma is the reason Lecture 1 came first: it turns every question about
continuity into a question about sequences, so every result about limits of
sums and products carries straight over.

\begin{theorem}[Extreme value theorem]
  A continuous function on a closed bounded interval $[a, b]$ attains a
  largest and a smallest value on that interval.
\end{theorem}

\begin{example}
  The function $f(x) = 1/x$ attains no largest value on the open interval
  $(0, 1)$. The interval is bounded but not closed, so the theorem does not
  apply, which shows the hypothesis is doing real work.
\end{example}

\begin{remark}
  Next lecture: uniform continuity, and why $f(x) = x^2$ is continuous
  everywhere but not uniformly continuous on the whole real line.
\end{remark}

\end{document}

How to use this template

Everything runs in your browser: nothing to install, no login, no compile timeout.

Click Open in LaTeX.to. The template opens as a project in the Projects pane, with main.tex in the editor and its finished PDF in the preview.
Replace the placeholders, such as Your Name, with your own text. The % comments in the files mark the places to edit.
Click Preview, or press Ctrl+Enter (Cmd+Enter on a Mac), to compile. The preview shows the new PDF; you do not need to save first.
If the compile fails, the Console pane opens with the LaTeX log. The first error in it says what went wrong and on which line.
To add an image, a .bib file or another .tex file, open the project's menu (the three dots next to its name in the Projects pane) and choose Upload file or New file.
The arrow next to Preview has Download PDF. Ctrl+S saves your edits, and the project stays in your browser on this device.

How to add a lecture

Each lecture is a \section, and the numbering of definitions and theorems restarts with it.

Add \section{Lecture 3: Differentiation} after the end of lecture 2 and write below it.
Definitions, theorems, lemmas, examples and remarks share one counter per section, so the third block in lecture 3 is 3.3 whatever kind it is.
The table of contents at the top lists the new lecture on the next compile.

How to keep one file per lecture

A term of notes gets long, and a file per lecture keeps main.tex short.

In the Projects pane, open the project's menu, choose New file and name it lecture-03.tex.
Put the \section and everything under it in that file, with no preamble and no \begin{document}.
In main.tex, put \input{lecture-03} where the lecture belongs, one line per file.
To compile a single lecture, put % in front of the other \input lines; the section numbers then start from 1.

How to state a theorem with a proof

The environments for the blocks are set up at the top of the file.

Write \begin{theorem}[Name of the theorem] and \end{theorem} around the statement; the name in brackets is optional.
Put the proof between \begin{proof} and \end{proof}, which ends it with the square.
Give the block a \label{thm:name} on its first line and refer to it with Theorem~\ref{thm:name}.

More templates: Exam and all templates.