Updated title page

chapters/real_analysis_2.tex

@@ -3,4 +3,6 @@
 
 \begin{document}
     \chapter{Real Analysis: Part II}
+
+    \subfile{sections/lims_and_funcs.tex}
 \end{document}

chapters/sections/lims_and_funcs.tex

@@ -0,0 +1,52 @@
+\documentclass[../script.tex]{subfiles}
+%! TEX root = ../../script.tex
+
+\begin{document}
+\section{Limits and Functions}
+
+In this chapter we will introduce the notation
+\[
+    \oball(x) = (x - \epsilon, x + \epsilon)
+\]
+
+\begin{defi}
+    Let $D \subset \realn$ and $x \in \realn$. $x$ is called a boundary point of $D$ if 
+    \[
+        \forall \epsilon > 0: ~~D \cap \oball(x) \ne 0
+    \]
+    The set of all boundary points of $D$ is called closure and is denoted as $\closure{D}$.
+\end{defi}
+
+\begin{eg}
+    \begin{enumerate}[(i)]
+        \item $x \in D$ is always a boundary point of $D$, because 
+        \[
+            x \in D \cap \oball(x)
+        \]
+
+        \item Boundary points don't have to be elements of $D$. If $D = (0, 1)$, then $0$ and $1$ are boundary points, because 
+        \[
+            \frac{\epsilon}{2} \in (0, 1) \cap \oball(0) = (-\epsilon, \epsilon) ~~\forall \epsilon > 0
+        \]
+
+        \item Let $D = \ratn$. Every $x \in \realn$ is a boundary point, because $\forall \epsilon > 0$, $\oball(x)$ contains at least one rational number. I.e. $\closure{\ratn} = \realn$.
+    \end{enumerate}
+\end{eg}
+
+\begin{rem}
+    If $x$ is a boundary point, then 
+    \[
+        \forall \epsilon > 0 ~\exists y \in D: ~~|x - y| < \epsilon 
+    \]
+    If $x$ is not a boundary point, then 
+    \[
+        \exists \epsilon > 0 ~\forall y \in D: ~~|x - y| \ge \epsilon 
+    \]
+\end{rem}
+
+\begin{thm}
+    \[
+        x \in \realn \text{ is a boundary point of } D \subset \realn \iff \exists \anyseqdef{D} \text{ such that } x_n \rightarrow x
+    \]
+\end{thm}
+\end{document}

chapters/sections/numbers.tex

@@ -385,7 +385,7 @@ $\intn$ are called integers, and $\ratn$ are called the rational numbers. $\natn
 The second statement implies that $\ratn$ is a field.
 \end{rem}
 
-\begin{cor}[Density of the rationals]
+\begin{cor}[Density of the rationals]\label{cor:densityrats}
 $x, y \in \realn, ~x < y$. Then
 \[
 	\exists r \in \ratn: ~~x < r < y

script.tex

@@ -54,9 +54,12 @@
 \newcommand{\rseqdef}[1]{\seq{#1} \subset \realn}
 \newcommand{\cseqdef}[1]{\seq{#1} \subset \cmpln}
 \newcommand{\rcseqdef}[1]{\seq{#1} \subset \realn \text{ (or } \cmpln \text{)}}
+\newcommand{\anyseqdef}[2][x]{\seq{#1} \subset #2}
 \newcommand{\series}[2][\infty]{\sum_{#2 = 1}^{#1}}
 \newcommand{\finite}{\text{ finite}}
 \newcommand{\conj}[1]{\overline{#1}}
+\newcommand{\oball}[1][\epsilon]{B_{#1}}
+\newcommand{\closure}[1]{\overline{#1}}
 
 \newcommand{\conv}[1]{\xrightarrow{\makebox[2em][c]{$\scriptstyle#1$}}}
 \newcommand{\convinf}{\conv{n \rightarrow \infty}}
@@ -111,8 +114,18 @@
 \begin{document}
 \title{Mathematics for Physicists}
 \author{\href{https://www.github.com/Lauchmelder23/Mathematics}{https://www.github.com/Lauchmelder23/Mathematics}}
-\maketitle
-\doclicenseThis
+
+\begin{titlepage}
+  \begin{center}
+    \vspace*{5cm}
+    \Huge{Mathematics for Physicists}\\[1cm]
+    \Large{https://www.github.com/Lauchmelder23/Mathematics}\\
+    \Large{Alma Mater Lipsiensis}\\[0,5cm]
+    \normalsize{\today} \\[7,5cm]
+  \end{center}
+  \doclicenseThis
+\end{titlepage}
+
 \tableofcontents
 
 \subfile{chapters/FaN.tex}