525 lines
20 KiB
Markdown
525 lines
20 KiB
Markdown
---
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title: Bounds
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TARGET DECK: Obsidian::STEM
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FILE TAGS: calculus::bounds
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tags:
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- calculus
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---
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## Overview
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Suppose $S$ is a nonempty set of real numbers and suppose there are numbers $L$ and $U$ such that $L \leq x \leq U$ for all $x \in S$. Then $S$ is said to be **bounded below** by $L$ and **bounded above** by $U$. The number $L$ is said to be a **lower bound** for $S$; the number $U$ is said to be an **upper bound** for $S$.
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%%ANKI
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Basic
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Let $\varnothing \subset S \subseteq \mathbb{R}$. What does it mean for $S$ to be bounded below by $B$?
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Back: For all $x \in S$, $B \leq x$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133063-->
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END%%
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%%ANKI
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Basic
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What does it mean for $\varnothing \subseteq \mathbb{R}$ to be bounded above by $B$?
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Back: N/A.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115335382-->
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END%%
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%%ANKI
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Basic
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Suppose $S \subseteq \mathbb{R}$ is bounded below by $B$. What kind of mathematical object is $S$?
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Back: A nonempty set.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115335387-->
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END%%
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%%ANKI
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Basic
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Suppose $S \subseteq \mathbb{R}$ is unbounded above. What kind of mathematical object is $S$?
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Back: A set (possibly empty).
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115335393-->
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END%%
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%%ANKI
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Basic
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Is the set of positive real numbers bounded below?
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Back: Yes.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133068-->
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END%%
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%%ANKI
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Basic
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Let $\varnothing \subset S \subseteq \mathbb{R}$. What does it mean for $S$ to be bounded above by $B$?
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Back: For all $x \in S$, $x \leq B$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133071-->
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END%%
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%%ANKI
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Basic
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Is the set of positive real numbers bounded above?
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Back: No.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133074-->
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END%%
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%%ANKI
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Basic
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What are the upper bounds of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: All real numbers $x \geq 1$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133077-->
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END%%
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%%ANKI
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Basic
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What are the upper bounds of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: All real numbers $x \geq 1$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133080-->
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END%%
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%%ANKI
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Basic
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What are the lower bounds of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: All real numbers $x \leq 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133084-->
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END%%
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%%ANKI
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Basic
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What are the lower bounds of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: All real numbers $x \leq 0$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133087-->
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END%%
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If $L \in S$, then $L$ is the **minimum element** of $S$ (denoted $L = \mathop{\text{min}} S$). Likewise, if $U \in S$, then $U$ is the **maximum element** of $S$ (denoted $U = \mathop{\text{max}}S$). A set with no lower bound is said to be **unbounded below**. A set with no upper bound is said to be **unbounded above**.
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%%ANKI
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Basic
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What is a maximum element of set $\varnothing \subset S \subseteq \mathbb{R}$?
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Back: A member of $S$ that is also an upper bound.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133090-->
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END%%
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%%ANKI
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Basic
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What is the maximum element of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: $1$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133094-->
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END%%
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%%ANKI
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Basic
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What is the maximum element of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: N/A.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133110-->
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END%%
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%%ANKI
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Basic
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What is a minimum element of set $\varnothing \subset S \subseteq \mathbb{R}$?
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Back: A member of $S$ that is also a lower bound.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133099-->
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END%%
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%%ANKI
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Basic
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What is the minimum element of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: N/A.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133103-->
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END%%
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%%ANKI
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Basic
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What is the minimum element of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: $0$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115133106-->
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END%%
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%%ANKI
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Basic
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How is a maximum element of $\varnothing \subset S \subseteq \mathbb{R}$ denoted?
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Back: As $\mathop{\text{max}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115335395-->
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END%%
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%%ANKI
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Basic
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How is a minimum element of $\varnothing \subset S \subseteq \mathbb{R}$ denoted?
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Back: As $\mathop{\text{min}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115335390-->
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END%%
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## Least Upper Bounds
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A number $B$ is called a **least upper bound** (or **supremum**) of a nonempty set $S$ if $B$ is an upper bound for $S$ and no number less than $B$ is an upper bound for $S$. This is denoted as $B = \mathop{\text{lub}}S$ or $B = \mathop{\text{sup}} S$.
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%%ANKI
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Basic
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Let $\varnothing \subset S \subseteq \mathbb{R}$. What is a least upper bound of $S$?
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Back: An upper bound $B$ for $S$ such that no number less than $B$ is also an upper bound for $S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953300-->
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END%%
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%%ANKI
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Basic
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What is an alternative term for a least upper bound of $S$?
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Back: A supremum of $S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953329-->
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END%%
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%%ANKI
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Basic
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How is a least upper bound for $S$ denoted?
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Back: As $\mathop{\text{lub}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953335-->
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END%%
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%%ANKI
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Basic
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How is a supremum for $S$ denoted?
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Back: As $\mathop{\text{sup}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953341-->
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END%%
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%%ANKI
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Basic
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What distinguishes a supremum from a least upper bound?
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Back: N/A. They are synonyms of one another.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953346-->
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END%%
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%%ANKI
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Basic
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What distinguishes a supremum from a maximum?
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Back: A supremum is not necessarily a member of the reference set.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953352-->
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END%%
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%%ANKI
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Basic
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What is the least upper bound of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: $1$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953358-->
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END%%
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%%ANKI
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Basic
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What is the supremum of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: $1$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953364-->
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END%%
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%%ANKI
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Basic
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How many upper bounds can a nonempty subset of $\mathbb{R}$ have?
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Back: $0$ or more.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953369-->
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END%%
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%%ANKI
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Basic
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How many supremums can a nonempty subset of $\mathbb{R}$ have?
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Back: $0$ or $1$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953375-->
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END%%
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%%ANKI
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Basic
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Is a maximum of a nonempty subset $S$ of $\mathbb{R}$ a supremum of $S$?
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Back: Yes.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953380-->
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END%%
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%%ANKI
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Basic
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Is a least upper bound of a nonempty subset $S$ of $\mathbb{R}$ a maximum of $S$?
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Back: Not necessarily.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953386-->
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END%%
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%%ANKI
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Basic
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Let $S \subseteq \mathbb{R}$ have a supremum. If $h > 0$, *why* does there exist an $x \in S$ such that $x > \mathop{\text{sup}} S - h$?
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Back: Otherwise $\mathop{\text{sup}}S - h$ is an upper bound less than $\mathop{\text{sup}}S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640370-->
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END%%
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%%ANKI
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Basic
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Let $S \subseteq \mathbb{R}$ have a supremum. If $h > 0$, *why* does there exist an $x \in S$ such that $x < \mathop{\text{sup}} S - h$?
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Back: N/A. This is not necessarily the case.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640373-->
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END%%
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%%ANKI
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Basic
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Let $S \subseteq \mathbb{R}$ have an infimum. If $h > 0$, *why* does there exist an $x \in S$ such that $x > \mathop{\text{inf}} S + h$?
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Back: N/A. This is not necessarily the case.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640374-->
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END%%
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%%ANKI
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Basic
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Let $S \subseteq \mathbb{R}$ have an infimum. If $h > 0$, *why* does there exist an $x \in S$ such that $x < \mathop{\text{inf}} S + h$?
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Back: Otherwise $\mathop{\text{inf}}S + h$ is a lower bound greater than $\mathop{\text{inf}}S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640375-->
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END%%
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%%ANKI
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Basic
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Let $A, B \subseteq \mathbb{R}$ have supremums. What set $C$ satisfies $\mathop{\text{sup}}C = \mathop{\text{sup}}A + \mathop{\text{sup}}B$?
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Back: $C = \{a + b \mid a \in A, b \in B\}$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640376-->
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END%%
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%%ANKI
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Basic
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Let $A, B \subseteq \mathbb{R}$. When is $\mathop{\text{sup}} \,\{a + b \mid a \in A, b \in B\}$ defined?
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Back: When $A$ and $B$ both have supremums.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640377-->
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END%%
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%%ANKI
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Basic
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Let $A, B \subseteq \mathbb{R}$. When is $\mathop{\text{inf}} \,\{a + b \mid a \in A, b \in B\}$ defined?
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Back: When $A$ and $B$ both have infimums.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640378-->
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END%%
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%%ANKI
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Basic
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Let $A, B \subseteq \mathbb{R}$ have infimums. What set $C$ satisfies $\mathop{\text{inf}}C = \mathop{\text{inf}}A + \mathop{\text{inf}}B$?
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Back: $C = \{a + b \mid a \in A, b \in B\}$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523640379-->
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END%%
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%%ANKI
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Basic
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Let $S, T \subseteq \mathbb{R}$ be nonempty sets such that $\forall s \in S, \forall t \in T, s \leq t$. Does $S$ have a supremum?
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Back: Yes.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523911490-->
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END%%
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%%ANKI
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Basic
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Let $S, T \subseteq \mathbb{R}$ be nonempty sets such that $\forall s \in S, \forall t \in T, s \leq t$. Does $T$ have a supremum?
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Back: Indeterminate.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523911494-->
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END%%
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%%ANKI
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Basic
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Let $S, T \subseteq \mathbb{R}$ be nonempty sets such that $\forall s \in S, \forall t \in T, s \leq t$. Does $S$ have an infimum?
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Back: Indeterminate.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523911496-->
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END%%
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%%ANKI
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Basic
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Let $S, T \subseteq \mathbb{R}$ be nonempty sets such that $\forall s \in S, \forall t \in T, s \leq t$. Does $T$ have an infimum?
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Back: Yes.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523911497-->
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END%%
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%%ANKI
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Basic
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Let $S, T \subseteq \mathbb{R}$ be nonempty sets such that $\forall s \in S, \forall t \in T, s \leq t$. How does $\mathop{\text{sup}} S$ compare to $\mathop{\text{inf}} T$?
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Back: $\mathop{\text{sup}}S \leq \mathop{\text{inf}}T$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724523911499-->
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END%%
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### Completeness Axiom
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Every nonempty set $S$ of real numbers which is bounded above has a supremum; that is, there is a real number $B$ such that $B = \mathop{\text{sup}} S$.
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%%ANKI
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Basic
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What does the completeness axiom of real numbers state?
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Back: Every nonempty set of real numbers bounded above has a supremum.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724116323038-->
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END%%
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%%ANKI
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Basic
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Consider $\varnothing \subseteq \mathbb{R}$. Why doesn't the completeness axiom of real numbers apply?
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Back: $\varnothing$ is not a nonempty set.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724116323044-->
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END%%
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%%ANKI
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Basic
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Consider $(0, 1) \subseteq \mathbb{R}$. Why doesn't the completeness axiom of real numbers apply?
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Back: N/A. It does.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724116323046-->
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END%%
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%%ANKI
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Basic
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Consider $\mathbb{R}^+$. Why doesn't the completeness axiom apply?
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Back: $\mathbb{R}^+$ is not bounded above.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724116323049-->
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END%%
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%%ANKI
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Basic
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Consider $(-\infty, 0)$. Why doesn't the completeness axiom apply?
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Back: N/A. It does.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724116323052-->
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END%%
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%%ANKI
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Basic
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What arbitrary decision was made when formulating the completeness axiom of real numbers?
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Back: Whether to use supremums or infimums.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%
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## Greatest Lower Bounds
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A number $B$ is called a **greatest lower bound** (or **infimum**) of a nonempty set $S$ if $B$ is a lower bound for $S$ and no number greater than $B$ is a lower bound for $S$. This is denoted as $B = \mathop{\text{glb}} S$ or $B = \mathop{\text{inf}} S$.
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%%ANKI
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Basic
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Let $\varnothing \subset S \subseteq \mathbb{R}$. What is a greatest lower bound of $S$?
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Back: A lower bound $B$ for $S$ such that no number greater than $B$ is also a lower bound for $S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%
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%%ANKI
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Basic
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What is an alternative term for a greatest lower bound of $S$?
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Back: An infimum of $S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953397-->
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END%%
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%%ANKI
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Basic
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How is a greatest lower bound for $S$ denoted?
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Back: As $\mathop{\text{glb}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953402-->
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END%%
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%%ANKI
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Basic
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How is an infimum for $S$ denoted?
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Back: As $\mathop{\text{inf}} S$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953408-->
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END%%
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%%ANKI
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Basic
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What distinguishes a greatest lower bound from an infimum?
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Back: N/A. They are synonyms of one another.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953417-->
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END%%
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%%ANKI
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Basic
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What distinguishes an infimum from a minimum?
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Back: An infimum is not necessarily a member of the reference set.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953422-->
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END%%
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%%ANKI
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Basic
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What is the infimum of interval $[0, 1] \subseteq \mathbb{R}$?
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Back: $0$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953428-->
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END%%
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%%ANKI
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Basic
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What is the greatest lower bound of interval $(0, 1) \subseteq \mathbb{R}$?
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Back: $0$
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953433-->
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END%%
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%%ANKI
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Basic
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How many lower bounds can a nonempty subset of $\mathbb{R}$ have?
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Back: $0$ or more.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953438-->
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END%%
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%%ANKI
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Basic
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How many greatest lower bounds can a nonempty subset of $\mathbb{R}$ have?
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Back: $0$ or $1$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953443-->
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END%%
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%%ANKI
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Basic
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Is an infimum of a nonempty subset $S$ of $\mathbb{R}$ a minimum of $S$?
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Back: Not necessarily.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953448-->
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END%%
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%%ANKI
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Basic
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Is a minimum of a nonempty subset $S$ of $\mathbb{R}$ a greatest lower bound of $S$?
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Back: Yes.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724115953452-->
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END%%
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## Bibliography
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* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980). |