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a439b45d2c
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| a439b45d2c | |||
| 6e76c14c70 | |||
| 78acee31ef |
18
Jenkinsfile
vendored
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18
Jenkinsfile
vendored
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pipeline {
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agent linux {
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stages {
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stage('Build') {
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steps {
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echo 'Starting build step...'
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sh './scripts/build.sh'
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}
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}
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// Testing latex isn't really a thing, but we could do basic sanity checks in the future?
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stage('Deploy') {
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echo 'Starting deploy step...'
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}
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}
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}
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}
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\subfile{week2/main.tex}
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\subfile{week2/main.tex}
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\clearpage
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\clearpage
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\subfile{week3/main.tex}
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\subfile{week3/main.tex}
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\clearpage
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\subfile{week4/main.tex}
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\end{document}
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\end{document}
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BIN
assignments/week4/main.pdf
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BIN
assignments/week4/main.pdf
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59
assignments/week4/main.tex
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59
assignments/week4/main.tex
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\documentclass[../main_text.tex]{subfiles}
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\begin{document}
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\setcounter{exercise}{0}
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\part{Assignment 4}
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\exercise*
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\begin{tcolorbox}
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We say a set $F \subset \R$ is \textit{closed} if its complement $F^c := \R \backslash F$ is open. Since
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$\emptyset$ and $\R$ are open, it follows that $\emptyset$ and $\R$ are closed as well.
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\begin{enumerate}[label=\emph{(\alph*)}]
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\item Let $a,b \in \R$ with $a < b$. Prove that $[a,b]$ is closed.
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\item Is the set $\Z \subset \R$ closed? Provide a proof to substantiate your claim.
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\item Is the set of rationals $\Q \subset \R$ closed? Provide a proof to substantiate your claim.
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\end{enumerate}
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\end{tcolorbox}
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\begin{enumerate}[label=\emph{(\alph*)}]
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\item The complement of $[a,b]$ is equal to the union of $(-\infty, a)$ and $(b, \infty)$. So, we have to prove
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that both these sets are open. But we have already done so in the assignment of the previous week, so I'll just
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leave it at that.
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\item To prove that $\Z \subset \R$ is closed, we have to prove that the complement is open. Since the complement
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consists of the union of a countably infinite number of open intervals $(a, b)$ such that $a < b$, we know
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from a combination of earlier exercises that this is the case. This is because any interval $(a, b)$ is open
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if $a,b \in \R$ such that $a < b$ and for any two open interval $A$ and $B$ that are open, then $A \cup B$
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is open as well.
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\item I claim that the set of rationals isn't closed in $\R$. This is because there doesn't exist any interval
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$(a,b)$ where $a,b \in \Q$ such that $a < b$, since for any $a$ and $b$ you can always find a $c$ such that
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$a < c < b$. This makes it impossible to find an $\epsilon > 0$ such that for any $x \in (a,b)$,
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$(x - \epsilon, x + \epsilon)$ is also in $(a,b)$ but in such a way that it only contains irrational numbers.
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This argument makes use of the fact that $\Q$ is dense in $\R$.
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\end{enumerate}
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\exercise*
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\begin{tcolorbox}
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\begin{enumerate}[label=\emph{(\alph*)}]
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\item Let $\Lambda$ be a set (not necessarily a subset of $\R$), and for each $\lambda \in \Lambda$, let
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$F_\lambda \in \R$. Prove that if $F_\lambda$ is closed for all $\lambda \in \Lambda$ then the set
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\begin{equation*}
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\bigcap_{\lambda \in \Lambda} F_\lambda = \{x \in \R : x \in F_\lambda \text{ for all }
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\lambda \in \Lambda\}
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\end{equation*}
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is closed.
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\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
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$\bigcup_{m=1}^n F_m$ is closed.
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\end{enumerate}
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\end{tcolorbox}
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This exercise is very similar to an exercise of the previous assignment, in which we looked at unions and intersections
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of \textit{open} intervals, whereas in this exercise, it's all about closed intervals. As the definition of closed
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intervals is intricately linked to the definition of open intervals, the following arguments will look very similar and
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shouldn't be surprising.
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\begin{enumerate}[label=\emph{(\alph*)}]
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\item
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\item
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\end{enumerate}
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\end{document}
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32
scripts/build.sh
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32
scripts/build.sh
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#!/bin/bash
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# Check if necessary packages are installed
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necessary_packages=("pdflatex")
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for package in ${necessary_packages}; do
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if ! [dpkg -l ${package} > /dev/null]; then
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echo "Error: ${package} is not installed."
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exit 1
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fi
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done
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# Build all directories
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subdirectory_file_name=main
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for D in *; do
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if [ -d "${D}" ]; then
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echo "Building ${D}..."
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cd ${D}
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pdflatex -interaction=nonstopmode -halt-on-error ${subdirectory_file_name}.tex
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cd ..
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fi
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done
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# Build main PDF
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main_file_name=main_text
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echo "Building main PDF..."
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pdflatex -interaction=nonstopmode -halt-on-error ${main_file_name}.tex
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# Clean up
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rm -rf *.aux *.log *.out *.toc
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