Compare commits
3 Commits
master
...
a439b45d2c
| Author | SHA1 | Date | |
|---|---|---|---|
| a439b45d2c | |||
| 6e76c14c70 | |||
| 78acee31ef |
18
Jenkinsfile
vendored
Normal file
18
Jenkinsfile
vendored
Normal file
@@ -0,0 +1,18 @@
|
||||
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...'
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
Binary file not shown.
@@ -17,4 +17,6 @@
|
||||
\subfile{week2/main.tex}
|
||||
\clearpage
|
||||
\subfile{week3/main.tex}
|
||||
\clearpage
|
||||
\subfile{week4/main.tex}
|
||||
\end{document}
|
||||
|
||||
BIN
assignments/week4/main.pdf
Normal file
BIN
assignments/week4/main.pdf
Normal file
Binary file not shown.
59
assignments/week4/main.tex
Normal file
59
assignments/week4/main.tex
Normal file
@@ -0,0 +1,59 @@
|
||||
\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}
|
||||
32
scripts/build.sh
Normal file
32
scripts/build.sh
Normal file
@@ -0,0 +1,32 @@
|
||||
#!/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
|
||||
Reference in New Issue
Block a user