Lecture Notes

What’s inside

• Starter files
• main.tex2.0 KB

  \documentclass[11pt,DIV=11]{scrartcl}
  
  \usepackage{amsmath,amssymb,amsthm}
  \usepackage{microtype}
  
  \theoremstyle{plain}
  \newtheorem{theorem}{Theorem}[section]
  \newtheorem{lemma}[theorem]{Lemma}
  \theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{example}[theorem]{Example}
  \theoremstyle{remark}
  \newtheorem*{remark}{Remark}
  
  \title{Lecture Notes: Real Analysis}
  \author{Your Name}
  \date{Fall 2026}
  
  \begin{document}
  
  \maketitle
  \tableofcontents
  
  \section{Lecture 1: Sequences and Limits}
  
  \begin{definition}[Convergence]
  A sequence $(a_n)$ of real numbers \emph{converges} to $L \in \mathbb{R}$
  if for every $\varepsilon > 0$ there exists $N$ such that
  $|a_n - L| < \varepsilon$ for all $n \geq N$. We write
  $\lim_{n \to \infty} a_n = L$.
  \end{definition}
  
  \begin{example}
  The sequence $a_n = 1/n$ converges to $0$: given $\varepsilon > 0$,
  take $N > 1/\varepsilon$.
  \end{example}
  
  \begin{theorem}[Uniqueness of limits]
  A convergent sequence has exactly one limit.
  \end{theorem}
  
  \begin{proof}
  Suppose $a_n \to L$ and $a_n \to M$. For any $\varepsilon > 0$, choose
  $n$ with $|a_n - L| < \varepsilon/2$ and $|a_n - M| < \varepsilon/2$;
  then $|L - M| < \varepsilon$. Since $\varepsilon$ was arbitrary, $L = M$.
  \end{proof}
  
  \begin{remark}
  Convergence in $\mathbb{R}$ is a special case of convergence in a metric
  space; we return to this in a later lecture.
  \end{remark}
  
  \section{Lecture 2: Series}
  
  \begin{definition}[Series]
  Given a sequence $(a_n)$, the \emph{series} $\sum_{n=1}^{\infty} a_n$
  converges if the sequence of partial sums $s_k = \sum_{n=1}^{k} a_n$
  converges.
  \end{definition}
  
  \begin{example}[Geometric series]
  For $|r| < 1$,
  \begin{equation}
    \sum_{n=0}^{\infty} r^n = \frac{1}{1 - r}.
  \end{equation}
  \end{example}
  
  \begin{lemma}[Divergence test]
  If $\sum a_n$ converges, then $a_n \to 0$. The converse fails: the
  harmonic series $\sum 1/n$ diverges although $1/n \to 0$.
  \end{lemma}
  
  % Add a section per lecture, or ask the agent in the chat alongside this
  % document to draft or typeset the next lecture for you.
  
  \end{document}
  

• Libraries and docs
• main.pdfprebuilt · 125 KBReplaced by the first recompile.

  Binary file, seeded as-is.

• main.synctex.gzprebuilt · 6.1 KBReplaced by the first recompile.

  Binary file, seeded as-is.