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Jenkinsfile vendored Normal file
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pipeline {
agent linux {
stages {
stage('Build') {
steps {
echo 'Starting build step...'
sh './scripts/build.sh'
}
}
// Testing latex isn't really a thing, but we could do basic sanity checks in the future?
stage('Deploy') {
echo 'Starting deploy step...'
}
}
}
}

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\subfile{week2/main.tex} \subfile{week2/main.tex}
\clearpage \clearpage
\subfile{week3/main.tex} \subfile{week3/main.tex}
\clearpage
\subfile{week4/main.tex}
\end{document} \end{document}

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assignments/week4/main.pdf Normal file

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\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
\item
\end{enumerate}
\end{document}

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scripts/build.sh Normal file
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#!/bin/bash
# Check if necessary packages are installed
necessary_packages=("pdflatex")
for package in ${necessary_packages}; do
if ! [dpkg -l ${package} > /dev/null]; then
echo "Error: ${package} is not installed."
exit 1
fi
done
# Build all directories
subdirectory_file_name=main
for D in *; do
if [ -d "${D}" ]; then
echo "Building ${D}..."
cd ${D}
pdflatex -interaction=nonstopmode -halt-on-error ${subdirectory_file_name}.tex
cd ..
fi
done
# Build main PDF
main_file_name=main_text
echo "Building main PDF..."
pdflatex -interaction=nonstopmode -halt-on-error ${main_file_name}.tex
# Clean up
rm -rf *.aux *.log *.out *.toc