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b/assignments/week2/main.tex index c4a2bd2..3dc243c 100644 --- a/assignments/week2/main.tex +++ b/assignments/week2/main.tex @@ -5,8 +5,6 @@ \exercise*[1.1.1] \begin{tcolorbox} - Prove: - Let $F$ be an ordered field and $x,y,z \in F$. If $x < 0$ and $y < z$, then $xy > xz$. \end{tcolorbox} @@ -19,4 +17,19 @@ Working out the right side with the distributive law, gives $0 < (-x*z)-(-x*y)$. Using $-1*-1 = 1$, gives $0 < (-xz)-(-xy)$, thus $0 < xy - xz$. The right part can be split again: $xz < xy$. Then, the < can be flipped, which gives $xy > xz$. \qed +\exercise[1.1.2] +\begin{tcolorbox} + Let $S$ be an ordered set. Let $A \subset S$ be a non-empty finite subset. Then A is bounded. Furthermore, + $\text{inf} A$ exists and in in $A$ and $\text{sup} A$ exists and is in $A$. +\end{tcolorbox} + +In order to prove that $A$ is bounded, we have to prove that it has an upper and a lower bound. Let us prove +that $A$ is bounded above first. + +In particular, we have to prove that $\exists a \in A$ such that $x \leq b$ for all $x \in E$. +Since $A$ is non-empty and finite, we can use induction on the cardinality of $A$, since that will always +be some natural number $n$. So, we have to prove two cases: the base case, where $|A|=1$, and the inductive step, +where we will assume that when $A$ has an upper bound when it has cardinality $m$, +then it also has an upper bound when its carindality is equal to $m+1$. + \end{document}

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