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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Let $\varnothing \subset S \subseteq \mathbb{R}$. What does it mean for $S$ to be bounded below by $B$?
Back: For all $x \in S$, $B \leq x$.
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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What does it mean for $\varnothing \subseteq \mathbb{R}$ to be bounded above by $B$?
Back: N/A.
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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Suppose $S \subseteq \mathbb{R}$ is bounded below by $B$. What kind of mathematical object is $S$?
Back: A nonempty set.
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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Suppose $S \subseteq \mathbb{R}$ is unbounded above. What kind of mathematical object is $S$?
Back: A set (possibly empty).
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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Is the set of positive real numbers bounded below?
Back: Yes.
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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Let $\varnothing \subset S \subseteq \mathbb{R}$. What does it mean for $S$ to be bounded above by $B$?
Back: For all $x \in S$, $x \leq B$.
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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Is the set of positive real numbers bounded above?
Back: No.
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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What are the upper bounds of interval $[0, 1] \subseteq \mathbb{R}$?
Back: All real numbers $x \geq 1$.
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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What are the upper bounds of interval $(0, 1) \subseteq \mathbb{R}$?
Back: All real numbers $x \geq 1$.
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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What are the lower bounds of interval $[0, 1] \subseteq \mathbb{R}$?
Back: All real numbers $x \leq 0$.
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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Basic
What are the lower bounds of interval $(0, 1) \subseteq \mathbb{R}$?
Back: All real numbers $x \leq 0$.
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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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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What is a maximum element of set $\varnothing \subset S \subseteq \mathbb{R}$?
Back: A member of $S$ that is also an upper bound.
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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What is the maximum element of interval $[0, 1] \subseteq \mathbb{R}$?
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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How is a maximum element of $\varnothing \subset S \subseteq \mathbb{R}$ denoted?
Back: As $\mathop{\text{max}} S$.
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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How is a minimum element of $\varnothing \subset S \subseteq \mathbb{R}$ denoted?
Back: As $\mathop{\text{min}} S$.
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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## Least Upper Bounds
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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Let $\varnothing \subset S \subseteq \mathbb{R}$. What is a least upper bound of $S$?
Back: An upper bound $B$ for $S$ such that no number less than $B$ is also an upper bound for $S$.
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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What is an alternative term for a least upper bound of $S$?
Back: A supremum of $S$.
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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How is a least upper bound for $S$ denoted?
Back: As $\mathop{\text{lub}} S$.
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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How is a supremum for $S$ denoted?
Back: As $\mathop{\text{sup}} S$.
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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What distinguishes a supremum from a least upper bound?
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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Let $A, B \subseteq \mathbb{R}$ have infimums. What set $C$ satisfies $\mathop{\text{inf}}C = \mathop{\text{inf}}A + \mathop{\text{inf}}B$?
Back: $C = \{a + b \mid a \in A, b \in B\}$
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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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?
Back: Yes.
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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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?
Back: Indeterminate.
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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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?
Back: Indeterminate.
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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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?
Back: Yes.
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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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 $\mathop{\text{inf}} T$?
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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What does the completeness axiom of real numbers state?
Back: Every nonempty set of real numbers bounded above has a supremum.
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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Consider $\varnothing \subseteq \mathbb{R}$. Why doesn't the completeness axiom of real numbers apply?
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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Consider $(-\infty, 0)$. Why doesn't the completeness axiom apply?
Back: N/A. It does.
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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What arbitrary decision was made when formulating the completeness axiom of real numbers?
Back: Whether to use supremums or infimums.
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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## Greatest Lower Bounds
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 $B = \mathop{\text{inf}} S$.
