From 6e76c14c70d60875870ebcaad4ef6ffc187f10bf Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Thu, 15 Jun 2023 17:03:22 +0200 Subject: [PATCH] Progress on assignment 4, first ones were amalgamations of earlier exercises --- assignments/main_text.pdf | Bin 637409 -> 639576 bytes assignments/week4/main.pdf | Bin 435169 -> 450514 bytes assignments/week4/main.tex | 36 ++++++++++++++++++++++++++++++++++-- 3 files changed, 34 insertions(+), 2 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index fee0e48416f9dfc427a45eb32fcfbe51a68247c5..83d3b602a28431283a7ec3b8ae34efe0d8f749f1 100644 GIT binary patch delta 13265 zcmajGV{|1zvo1W@;f`%P6Wg|JO>CPx#>BQJnAo<>iIa(KTX)`b&v(yS_x|`+tzP|9 zb#?czUQbtb_0ApTN(dDyg_4*AJu?G4EXC}?&IkDu8%RCxrNl1XV1u`4Qj+ZNuKhVUe@&ILp2k&vnpr6B3E;-8D9`Iigg1ehobn zdL-P5ZUA*taJ@4v_*VOXPw3~(ZStk#y{LUpszGZv%l&&8SK4i(fS(|1zIdOazZZ6I z{mL&7b#ZkzGqU>+ zaxk`rWoG4QxCaO$0oYlR`e@J^WKgG&5m;dv<;?6YTrG*2IarhGLcfD_*6r~)k^Po6 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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 + \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}