Problem Set - Orders on Z, Q, R and C

## Orders on $ℤ,ℚ,ℝ,$ and $ℂ$
 Define the order $\ge$ on ${ℤ}_{>0}$. Define the order $\ge$ on ${ℤ}_{\ge 0}$. Define the order $\ge$ on $ℤ$. Define the order $\ge$ on $ℚ$. Show that $\frac{a}{b}\le \frac{c}{d}$ if and only if $ab{d}^{2}\le cd{b}^{2}$. Define the order $\ge$ on $ℝ$. Show that there is no order $\ge$ on $ℂ$ such that $ℂ$ is a totally ordered field. Show that if $x,y,z\in ℝ$ and $x\le y$ and $y\le z$ then $x\le z$. Show that if $x,y\in ℝ$ and $x\le y$ and $y\le x$ then $x=y$. Show that if $x,y,z\in ℝ$ and $x\le y$ then $x+z\le y+z$. Show that if $x,y\in ℝ$ and $x\ge 0$ and $y\ge 0$ then $xy\ge 0$. Show that if $x\in ℝ-\left\{0\right\}$ then ${x}^{2}>0$. Show that if $x,y\in ℝ$ and $0 then ${y}^{-1}<{x}^{-1}$. (The Archimedean property of $ℝ$) Show that if $x,y\in ℝ$ and $x\in {ℝ}_{>0}$ then there exists $n\in {ℤ}_{\ge 0}$ such that $nx>y$. Show that the Archimedean property is equivalent to ${ℤ}_{>0}$ is an unbounded subset of $ℝ$. ($ℚ$ is dense $ℝ$) Show that if $x,y\in ℝ$ and $x then there exists $p\in ℚ$ such that $x. ($ℝ-ℚ$ is dense $ℝ$) Show that if $x,y\in ℝ$ and $x then there exists $p\in ℝ-ℚ$ such that $x. If $x,y\in ℝ$ and $x show that there exist infinitely many rational numbers between $x$ and $y$ as well as infinitely many irrational numbers. Let $x\in {ℝ}_{>0}$ and $n\in {ℤ}_{>0}$. Then there exists a unique $y\in {ℝ}_{>0}$ such that ${y}^{n}=x$. Find the minimal $N\in {ℤ}_{>0}$ such that $n<{2}^{n}$ for all $n\ge N$. Find the minimal $N\in {ℤ}_{>0}$ such that $n!>{2}^{n}$ for all $n\ge N$. Find the minimal $N\in {ℤ}_{>0}$ such that ${2}^{n}>2{n}^{3}$ for all $n\ge N$. For each of the following subsets of $ℝ$ find the maximum, the minimum, an upper bound, a lower bound, the supremum, and the infimum: (a)   $A=\left\{p\in ℚ|{p}^{2}<2\right\}$, (b)   $B=\left\{p\in ℚ|{p}^{2}>2\right\}$, (c)   ${E}_{1}=\left\{r\in ℚ|r<0\right\}$, (d)   ${E}_{2}=\left\{r\in ℚ|r\le 0\right\}$, (e)   $E=\left\{\frac{1}{n}|n\in {ℤ}_{>0}\right\}$, (f)   $\left[0,1\right)$, (g)   ${ℤ}_{>0}$, (h)   $\left\{x\in ℚ|x\le 0 \text{or} \left(x>0 \text{and} {x}^{2}>2\right)\right\}$, (i)   $ℤ$, (j)   $\left[\sqrt{2},2\right]$, (k)   $\left(\sqrt{2},2\right)$, (l)   $\left\{x\in ℝ|x=\frac{{\left(-1\right)}^{n}}{n}, n\in {ℤ}_{>0}\right\}$, (m)   $\left\{\frac{1}{{\left(|n|+1\right)}^{2}}|n\in ℤ\right\}$, (n)   $\left\{n+\frac{1}{n}|n\in {ℤ}_{>0}\right\}$, (o)   $\left\{{2}^{-m}-{3}^{n}|m,n\in {ℤ}_{\ge 0}\right\}$, (p)   $\left\{x\in ℝ|{x}^{3}-4x<0\right\}$, (q)   $\left\{1+{x}^{2}|x\in ℝ\right\}$, Let $S$ be a nonempty subset of $ℝ$. Show that $x=\mathrm{sup}S$ if and only if (a)   $x$ is an upper bound of $S$, and (b)   for every $\epsilon \in {ℝ}_{>0}$ there exists $y\in S$ such that $x-\epsilon . State and prove a characterization of $\mathrm{inf}S$ analogous to the characterization of $\mathrm{sup}S$ in the previous problem. Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that if $S$ is bounded then $c+S=\left\{c+s\text{\hspace{0.17em}}|\text{\hspace{0.17em}}s\in ℝ\right\}$ is bounded. Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that if $S$ is bounded then $cS=\left\{cs\text{\hspace{0.17em}}|\text{\hspace{0.17em}}s\in ℝ\right\}$ is bounded. Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that $\mathrm{sup}\left(c+S\right)=c+\mathrm{sup}S$. Let $c\in {ℝ}_{\ge 0}$ and let $S$ be a subset of $ℝ$. Show that $\mathrm{sup}\left(cS\right)=c\mathrm{sup}S$. Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that $\mathrm{inf}\left(c+S\right)=c+\mathrm{inf}S$. Let $c\in {ℝ}_{\le 0}$ and let $S$ be a subset of $ℝ$. Show that $\mathrm{inf}\left(cS\right)=c\mathrm{inf}S$.