Week 4 exercise 2 done by referring to earlier exercises

assignments/week4/main.tex

@@ -52,8 +52,24 @@ of \textit{open} intervals, whereas in this exercise, it's all about closed inte
 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
+    \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}
 
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