MATH 4630‑01
Homework Assignment #1
DUE: Friday, Jan 30, 11:59p
Sections 1.1–1.5
Instructions: Show all work clearly and justify each conclusion.
Collaboration is encouraged, but write up your solutions individually in your own words.
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
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Let $A, B \subseteq \mathbb{R}$ be nonempty and bounded.
- Prove that if $A \subseteq B$, then $\sup A \le \sup B$.
- Prove or disprove: If $\sup A < \inf B$, then there exists a real number $c$ such that $a < c < b$ for all $a \in A$ and $b \in B$.
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Let $a,b \in \mathbb{R}$. Recall the triangle inequality: $|x+y| \le |x| + |y|$ for all $x,y \in \mathbb{R}$.
- Use the identity $a = (a-b) + b$ to obtain an upper bound for $|a|$ in terms of $|a-b|$ and $|b|$.
- Interchange the roles of $a$ and $b$ to obtain a similar inequality for $|b|$.
- Deduce that $\bigl||a| - |b|\bigr| \le |a-b|$.
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Let $E$ and $F$ be nonempty subsets of $\mathbb{R}$ bounded above, and define $E + F = \{\, x + y : x \in E,\, y \in F \,\}$.
- Let $\alpha = \sup E$ and $\beta = \sup F$. Show that $\alpha + \beta$ is an upper bound for $E + F$.
- Let $M$ be an arbitrary upper bound for $E + F$, and fix $x \in E$. Show that $\beta \le M - x$.
- Conclude that $\sup(E + F) = \alpha + \beta$.
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For each set below, determine and justify $\sup A$ and $\inf A$, and state whether they belong to the set.
- $A = \{\,x \in \mathbb{R} : x^2 < 2\,\}$
- $A = \{\,1 - 1/n : n \in \mathbb{N}\,\}$
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Recall that a set is countable if it is finite or in one-to-one correspondence with $\mathbb{N}$.
- Prove that any finite subset of $\mathbb{R}$ is countable.
- Show that $A = \{\, 1/n : n \in \mathbb{N} \,\}$ is countable by explicitly describing a bijection with $\mathbb{N}$.
- Let $B = \mathbb{Z} \cap [0,\infty)$. Prove that $B$ is countable.
- Prove that the union of a finite set and a countable set is countable.
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