A Suprema and Infima

This section of the appendix contains an extremely brief review of basic facts concerning sets, functions, suprema, and infima. We recommend Bartle & Sherbert (2011) for those who wish to learn more.

A.1Sets and Functions¶

A set is a collection of objections viewed as a whole. Examples include the set of natural numbers $\NN \coloneq \{1, 2, \ldots\}$ and $\natset{n} \coloneq \{1, 2, \ldots, n\}$ when $n \in \NN$. The set that contains no elements is called the empty set and denoted by $\emptyset$.

Let $A$ and $B$ be two sets and let $A \times B$ be their Cartesian product, defined as the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$. A binary relation $\sim$ between two sets $A$ and $B$ is a subset of $A \times B$. If $(a, b)$ is in this subset we write $a \sim b$. An equivalence relation on $A$ is a binary relation $\sim$ between $A$ and itself that is reflexive, symmetric, and transitive. That is,

1. $a \sim a$ for all $a \in A$,

2. $a \sim a'$ implies $a' \sim a$, and

3. $a \sim a'$ and $a' \sim a''$ implies $a \sim a''$.

A function $f$ from set $A$ to set $B$, written $A \ni x \mapsto f(x) \in B$ or $f \colon A \to B$, is a rule (in fact, a binary relation) associating to each and every element $a$ in $A$ one and only one element $b \in B$. The point $b$ is also written as $f(a)$, and called the image of $a$ under $f$. For $C \subset A$, the set $f(C)$ is the set of all images of points in $C$, and is called the image of $C$ under $f$. Also, for $D \subset B$, the set $f^{-1}(D)$ is all points in $A$ that map into $D$ under $f$, and is called the preimage of $D$ under $f$.

A function $f \colon A \to B$ is called one-to-one if distinct elements of $A$ are always mapped into distinct elements of $B$, and onto if every element of $B$ is the image under $f$ of at least one point in $A$. A bijection or one-to-one correspondence from $A$ to $B$ is a function $f$ from $A$ to $B$ that is both one-to-one and onto.

A set $\Xsf$ is called finite if there exists a bijection from $\Xsf$ to $[n] \coloneq \{1, \ldots, n\}$ for some $n \in \NN$. In this case, we can write $\Xsf = \{x_1, \ldots, x_n\}$. The number $n$ is called the cardinality of $\Xsf$. Note that, according to our definition, every finite set is automatically nonempty.

If $f \colon A \to B$ and $g \colon B \to C$, then the composition of $f$ and $g$ is the function $g \circ f$ from $A$ to $C$ defined at $a \in A$ by $(g \circ f)(a) \coloneq g(f(a))$.

A.2Some Properties of The Real Line¶

Given a subset $A$ of $\RR$, we call $u \in \RR$ an upper bound of $A$ if $a \leq u$ for all $a$ in $A$. A lower bound of $A$ is any number $\ell$ such that $\ell \leq a$ for all $a \in A$. If $A$ has both an upper and lower bound then $A$ is called bounded. Equivalently, $A$ is bounded whenever there exists an $n \in \NN$ with $A \subset [-n, n]$.

Let $U(A)$ be the set of all upper bounds of $A$. An element $\bar u$ of $\RR$ is called a supremum or least upper bound of $A$ if

1. $\bar u \in U(A)$ and

2. $\bar u \leq u$ for every $u \in U(A)$.

When a supremum of $A$ exists in $\RR$, we write it as $\sup A$.

One of the most important properties of $\RR$ is stated below.

Theorem A.2.1 is often taken as axiomatic in formal constructions of the real numbers. (Alternatively, one may assume completeness of the reals and then prove Theorem A.2.1 using this property. See, e.g., Bartle & Sherbert (2011).)

If $i \in \RR$ is a lower bound for $A$ and also satisfies $i \geq \ell$ for every lower bound $\ell$ of $A$, then $i$ is called the infimum of $A$ and we write $i = \inf A$. At most one such $i$ exists, and every nonempty subset of $\RR$ bounded from below has an infimum.

A real sequence is a map $x$ from $\NN$ to $\RR$, with the value of the function at $k \in \NN$ typically denoted by $x_k$ rather than $x(k)$. A real sequence $x = (x_k)_{k \geq 1} \coloneq (x_k)_{k \in \NN}$ is said to converge to $\bar x \in \RR$ if, for each $\epsilon > 0$, there exists an $N \in \NN$ such that $|x_k - \bar x| < \epsilon$ for all $k \geq N$. In this case, we write $\lim_k x_k = \bar x$ or $x_k \to \bar x$. Bartle & Sherbert (2011) give an excellent introduction to real sequences and their basic properties.

A real sequence $(x_k)_{k \geq 1}$ is called increasing if $x_k \leq x_{k+1}$ for all $k$ and decreasing if $x_{k+1} \leq x_k$ for all $k$. If $(x_k)_{k \geq 1}$ is increasing (resp., decreasing) and $x_k \to x \in \RR$ then we also write $x_k \uparrow x$ (resp., $x_k \downarrow x$).

Let $(x_k)$ be a real sequence in $\RR$ and set $s_n \coloneq \sum_{k=1}^n x_k$. If the sequence $(s_n)$ converges to some $s \in \RR$, then we set

We say that the series $\sum_{k=1}^n x_k$ converges to $\sum_{k=1}^\infty x_k$.

A.3Max and Min¶

A number $m$ contained in a subset $A$ of $\RR$ is called the maximum of $A$ and we write $m = \max A$ if $a \leq m$ for every $a \in A$. It is called the minimum of $A$ and we write $m = \min A$ if $a \geq m$ for every $a \in A$.

A subset $A$ of $\RR$ is called closed if, for any sequence $(x_n)$ contained in $A$ and converging to some limit $x \in \RR$, the limit $x$ is in $A$.

Given an arbitrary set $D$ and a function $f \colon D \to \RR$, define

whenever the latter exists. The terms $\inf_{x \in D} f(x)$ and $\min_{x \in D} f(x)$ are defined analogously. A point $x^* \in D$ is called a

• maximizer of $f$ on $D$ if $x^* \in D$ and $f(x^*) \geq f(x)$ for all $x \in D$, and a

• minimizer of $f$ on $D$ if $x^* \in D$ and $f(x^*) \leq f(x)$ for all $x \in D$.

Equivalently, $x^* \in D$ is a maximizer of $f$ on $D$ if $f(x^*) = \max_{x \in D} f(x)$, and a minimizer if $f(x^*) = \min_{x \in D} f(x)$. We define

The set $\argmin_{x \in D} f(x)$ is defined analogously.