Week 4 exercise 2 done by referring to earlier exercises

.gitmodules

@@ -1,4 +1,4 @@
 [submodule "latex-homework"]
-	path = src/template
+	path = assignments/template
 	url = https://github.com/gijs-pennings/latex-homework.git
 	branch = master

Dockerfile

@@ -1,2 +0,0 @@
-FROM kjarosh/latex:2025.1
-RUN tlmgr install libertinus

Jenkinsfile

@@ -1,11 +1,10 @@
 pipeline {
-    agent any
+    agent {label 'linux'}
     stages {
         stage('Build') {
-            agent { dockerfile true }
             steps {
                 echo 'Starting build step...'
-                sh 'latexmk -pdf -outdir=out main_text.tex'
+                sh './scripts/build.sh'
             }
         }
 

assignments/main_text.tex

@@ -1,4 +1,4 @@
-\documentclass{src/template/homework}
+\documentclass{template/homework}
 
 \usepackage{enumitem} % Gives access to better enumeration items
 \usepackage{tcolorbox} % Gives boxes
@@ -6,15 +6,17 @@
 
 \title{MIT OCW Real Analysis}
 \author{Philippe H. Zwietering}
-\date{\today}
+\date{}
 
 \begin{document}
 \maketitle
 \tableofcontents
 \clearpage
-\subfile{src/week1.tex}
+\subfile{week1/main.tex}
 \clearpage
-\subfile{src/week2.tex}
+\subfile{week2/main.tex}
 \clearpage
-\subfile{src/week3.tex}
+\subfile{week3/main.tex}
+\clearpage
+\subfile{week4/main.tex}
 \end{document}

assignments/week4/main.tex

@@ -0,0 +1,75 @@
+\documentclass[../main_text.tex]{subfiles}
+\begin{document}
+\setcounter{exercise}{0}
+
+\part{Assignment 4}
+
+\exercise*
+\begin{tcolorbox}
+    We say a set $F \subset \R$ is \textit{closed} if its complement $F^c := \R \backslash F$ is open. Since
+    $\emptyset$ and $\R$ are open, it follows that $\emptyset$ and $\R$ are closed as well.
+
+    \begin{enumerate}[label=\emph{(\alph*)}]
+        \item Let $a,b \in \R$ with $a < b$. Prove that $[a,b]$ is closed.
+        \item Is the set $\Z \subset \R$ closed? Provide a proof to substantiate your claim.
+        \item Is the set of rationals $\Q \subset \R$ closed? Provide a proof to substantiate your claim.
+    \end{enumerate}
+\end{tcolorbox}
+
+\begin{enumerate}[label=\emph{(\alph*)}]
+    \item The complement of $[a,b]$ is equal to the union of $(-\infty, a)$ and $(b, \infty)$. So, we have to prove
+        that both these sets are open. But we have already done so in the assignment of the previous week, so I'll just
+        leave it at that.
+    \item To prove that $\Z \subset \R$ is closed, we have to prove that the complement is open. Since the complement
+        consists of the union of a countably infinite number of open intervals $(a, b)$ such that $a < b$, we know
+        from a combination of earlier exercises that this is the case. This is because any interval $(a, b)$ is open
+        if $a,b \in \R$ such that $a < b$ and for any two open interval $A$ and $B$ that are open, then $A \cup B$
+        is open as well.
+    \item I claim that the set of rationals isn't closed in $\R$. This is because there doesn't exist any interval
+        $(a,b)$ where $a,b \in \Q$ such that $a < b$, since for any $a$ and $b$ you can always find a $c$ such that
+        $a < c < b$. This makes it impossible to find an $\epsilon > 0$ such that for any $x \in (a,b)$,
+        $(x - \epsilon, x + \epsilon)$ is also in $(a,b)$ but in such a way that it only contains irrational numbers.
+        This argument makes use of the fact that $\Q$ is dense in $\R$.
+\end{enumerate}
+
+\exercise*
+\begin{tcolorbox}
+    \begin{enumerate}[label=\emph{(\alph*)}]
+        \item Let $\Lambda$ be a set (not necessarily a subset of $\R$), and for each $\lambda \in \Lambda$, let
+            $F_\lambda \in \R$. Prove that if $F_\lambda$ is closed for all $\lambda \in \Lambda$ then the set
+            \begin{equation*}
+                \bigcap_{\lambda \in \Lambda} F_\lambda = \{x \in \R : x \in F_\lambda \text{ for all }
+                \lambda \in \Lambda\}
+            \end{equation*}
+            is closed.
+        \item Let $n \in \N$, and let $F_1,...,F_n \subset \R$. Prove that if $F_1,...,F_n$ are closed then the set
+            $\bigcup_{m=1}^n F_m$ is closed.
+    \end{enumerate}
+\end{tcolorbox}
+
+This exercise is very similar to an exercise of the previous assignment, in which we looked at unions and intersections
+of \textit{open} intervals, whereas in this exercise, it's all about closed intervals. As the definition of closed
+intervals is intricately linked to the definition of open intervals, the following arguments will look very similar and
+shouldn't be surprising.
+\begin{enumerate}[label=\emph{(\alph*)}]
+    \item Non-formally speaking, in this exercise we want to prove that any intersection of closed sets in $\R$, is
+        closed itself. In order to make this formal, we will assume that $F_\lambda$ is closed for all $\lambda \in
+        \Lambda$ and follow the definition as presented.
+
+        So, let us assume that $F_\lambda$ is closed for all $\lambda \in \Lambda$. Following the definition of closed
+        sets, this means that the complement of $F_\lambda$ is open. Let's call the complement $\R \backslash F_\lambda =
+        U_\lambda$. Similarly, proving $\bigcap_{\lambda \in \Lambda} F_\lambda$ is closed, means we need to prove its
+        complement is open. From De Morgan's laws, it follows that $\R \backslash \bigcap_{\lambda \in \Lambda}
+        F_\lambda = \bigcup_{\lambda \in \Lambda} U_\lambda$.
+
+        We already proved this statement in exercise 5.b of week 3.
+    \item Similarly to the previous exercise, non-formally speaking we want to prove that any union of closed sets in
+        $\R$ is closed itself. Again, we will assign the complement of $(F_n)^c = U_n$, since we can't directly prove
+        a set is closed; we can only use the definition of closedness by proving something on open sets.
+
+        In the same vein, in order to prove that $\bigcup_{m=1}^n F_m$ is closed, we can only try and prove that its
+        complement is open. Using De Morgan's laws, $(\bigcup_{m=1}^n F_m)^c = \bigcap_{m=1}^n U_m$. This is the same
+        exercise as exercise 5.c from week 3, which also already proves the theorem.
+\end{enumerate}
+
+\end{document}