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Due: September 11th, 2020
Math 104 Assignment 2
Some useful facts
Prove the following useful facts.
- If $T \subseteq \Z$ is non-empty and bounded above in $\R$, then $\sup T \in T$. (In particular, $\sup T \in \Z$.)
Suppose that $\emptyset \subsetneq E_1 \subseteq E_2 \subseteq \R$.
If $E_2$ is bounded above, then $\sup E_1$ exists and $\sup E_1 \leq \sup E_2$.
(Note: the symbol "$\subsetneq$" means "is a proper subset of", so $A \subsetneq B$ means $A \subseteq B$ and $A \neq B$.
Thus $\emptyset \subsetneq E_1$ means "$E_1$ is non-empty". In contrast, the symbol $\nsubseteq$ means "is not a subset of". The symbol "$\subset$" is inconsistently used, usually meaning $\subsetneq$ but sometimes meaning $\subseteq$, depending on the author.)
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If $x, y \in \R$ with $0 \leq x \leq y$ then $x^2 \leq y^2$.
Moreover, assuming that $x^{1/2}, y^{1/2}$ exist in $\R$ with $0 \leq x^{1/2}, y^{1/2}$, prove that $x^{1/2}\leq y^{1/2}$.
(We will soon see in lecture that $x^{1/2}$ and $y^{1/2}$ do indeed exist in $\R$, but we aren't there yet.)
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If $S$ is an ordered set, $E, F \subseteq S$ have suprema in $S$, and they have the property that for every $e \in E$ there is $f \in F$ with $e \preceq f$, then $\sup E \preceq \sup F$.
-
Some facts about suprema
- Let $(S, \preceq)$ be an ordered set, and $t \in S$.
Prove that $\sup\set{s \in S \mid s \preceq t} = t$.
- Show by example that Part A need not hold if the $\preceq$ in the definition of the set is replaced by $\prec$.
- Show that, nonetheless, if $x \in \R$ then
$$\sup\set{q \in \Q \mid q \lt x} = x.$$
(Note that, as a result, every real number is the supremum of some set of rational numbers.)
Suprema in the rationals are suprema in the reals
- Suppose that $E \subset \Q$ has least upper bound $t \in \Q$.
Show that $t$ is also the least upper bound of $E \subset \R$.
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Next week we will prove that an irrational number exists. Use this to prove that $\Q$ does not have the Least Upper Bound Property.
Properties of irrational numbers
- Prove that if $q \in \Q$ and $t \in \R\setminus \Q$, then $q + t \in \R\setminus\Q$.
- Prove that if $q \in \Q$ and $t \in \R\setminus \Q$, then $qt \in \set{0}\cup(\R\setminus\Q)$.
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Prove that if $x, y \in \R$ with $x \lt y$, there is some $t \in \R\setminus\Q$ with $x \lt t \lt y$.
To do so, you may use the fact that an irrational number exists; if it is convenient, you may use in particular the fact that $\sqrt2\in\R$.
Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.
They are coloured by approximate difficulty: easy/medium/hard.
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Variations on a theme from lecture
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Suppose that $(\F, \leq)$ is an ordered field, $E \subseteq \F$, and $\alpha \in \F$.
Prove that $t \in \F$ is a lower bound for $E$ if and only if $t + \alpha$ is a lower bound for $E + \alpha$.
Consequently, prove that $\inf E$ exists if and only if $\inf(E+\alpha)$ exists, in which case $(\inf E)+\alpha = \inf(E+\alpha)$.
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Formulate and prove a similar statement about upper bounds and suprema of sets, when multiplied by some $t \in \F$ with $t \gt 0$.
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Formulate and prove a similar statement when $t \lt 0$.
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Prove that $\Z$ is not bounded below in $\R$.
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Prove that $\N$ is not bounded above in $\R$.
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Ordered fields
- Prove that an ordered field cannot be bounded above.
- Prove that in any ordered field $(\F, \leq)$, $\inf\set{x \in \F \mid x \gt 0} = 0$.
- Prove that if $\F$ is an ordered field and $N \in \N$,
$$\sum_{i=1}^N 1_\F = \underbrace{1_\F + 1_\F + \ldots + 1_\F}_{N \text{ times}} \neq 0.$$
(This is not true about fields in general. In fact, for any prime number $p$, there are fields within which
$$\sum_{i=1}^p 1_\F = 0.$$
Such fields are said to be "of characteristic $p$".
A field where $\sum_{i=1}^N 1_\F \neq 0$ for all $N \in \N$ is said to be "of characteristic $0$"; so $\R$, $\Q$, and $\C$ are all of characteristic $0$.
See, e.g., Wikipedia for more information.)
- Prove that an ordered field cannot be finite.