Finished first exercise of assignment 3 and did some refactoring on numbering etc

assignments/week3/main.tex

@@ -1,8 +1,34 @@
 \documentclass[../main_text.tex]{subfiles}
 \begin{document}
+\setcounter{exercise}{0}
 
-\section{Assignment 3}
+\part{Assignment 3}
 
 \exercise*
+\begin{tcolorbox}
+    Suppose $x,y \in \R$ and $x < y$. Prove that there exists $i \in \R\backslash\Q$ such that $x < i < y$.
+\end{tcolorbox}
+
+If either $x$ or $y$ (or both) are not rational numbers, we can simply take the average like so: $\frac{x+y}{2}$,
+in a similar way we did for the rationals. Since $x$ or $y$ isn't rational, the resulting fraction will also not be
+a rational and this proves the statement.
+
+Now if $x,y \in \Q$, we cannot use this average trick, because the resulting fraction will be a rational itself and
+so it doesn't satisfy the restriction that it must be in $\R\backslash\Q$. So we have to take a different approach.
+
+Let $x,y \in \Q$ with $x < y$ and $m := \frac{x+y}{2}$, so $x < m < y$. Then, let $X = \{a \in \R : x < a < m\}$ and
+let $Y = \{b \in \R : m < b < y\}$. Since $x < m$ and $m < y$, these are nonempty and they are bounded, because
+of the restrictions $x < a < m$ and $m < b < y$. So, there exists $k \in X$, that is not rational such that
+$x < k < m$ and there exists $h \in Y$ that is not rational such that $m < h < y$. Pick either $k$ or $h$ as $i$,
+since $x < k < m < h < y$. \qed
+
+\exercise*
+\begin{tcolorbox}
+    Let $E \subset (0,1)$ be the set of all real numbers with decimal representation using only the digits 1 and 2:
+    \begin{equation}
+        E := \{x \in (0,1) : \forall j \in \N, \exists d_j \in \{1,2\} \text{ such that } x = 0.d_1d_2...\}
+    \end{equation}
+    Prove that $|E| = |\mathcal{P}(\N)|$.
+\end{tcolorbox}
 
 \end{document}