Chapter Review

Summary

Finite sums obey the rules of Lemma 1.1.3 in every commutative ring, and a sum whose terms are differences telescopes (Theorem 1.1.7). The binomial theorem holds in every commutative ring with coefficients defined by Pascal’s rule (Theorem 1.2.3). In the real numbers it bounds a power from below by one or two of its terms (Corollary 1.2.4). Bernoulli’s inequality becomes strict (Lemma 1.3.1) and extends to products, and the arithmetic mean bounds the geometric mean (Theorem 1.3.3). With these, geometric sequences, $n/r^n$ and $1/n!$ tend to zero. Every non-negative real has a unique non-negative $n$ -th root (Lemma 1.5.1). The powers $a^{m/n}$ of a positive real are well defined and obey the laws of powers (Proposition 1.5.5).

What is open: powers with irrational exponents, which Chapter 8 defines through the exponential function, and the limit of the compound-interest sequence, which Analysis on the Real Line, Chapter 4 identifies.

Proof methods in this chapter

Telescoping (Example (Method: telescoping)). When a sum has terms $b_{k+1} - b_k$ , the middle cancels and only the ends survive. Introduced in Section 1.1, where it sums reciprocals and squares and proves the difference of powers. Its “verification form” checks a guessed closed form instead of finding it.
Bound a product by a power (Example (Method: bound a product by a power)). Replace the factors by their arithmetic mean, which is easy to compute, and let the arithmetic–geometric mean inequality turn the product into a power. Introduced in Section 1.3.
Keep the dominant term (Example (Method: keep the dominant term)). In a quotient of sums, keep the term that grows fastest and absorb the others into it from some index on. The quotient is then bounded by a quotient of single powers. Introduced in Section 1.4.
Bound a power from below (Example (Method: bound a power from below)). Write a base above one as $1 + h$ and keep one term of the expansion, by Bernoulli or by the binomial theorem. This makes $r^n$ at least a multiple of a power of $n$ . Introduced in Section 1.4, with exercises B1 and C2.
Recalled from earlier volumes. Proving a limit from the definition (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Example (Method: proving a limit from the definition)); induction on the last term (Sets and Functions, Example (Method: finding $P(n)$ inside $P(n + 1)$ )); well-definedness (Sets and Functions, Example (Method: checking that a definition on classes is well defined)). Constructing a point as a supremum, from Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Chapter 6 (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 6.1.4), is run again in Section 1.5.

Terms

arithmetic mean, geometric mean: $(a_1 + \dots + a_n)/n$ and $\sqrt[n]{a_1 \cdots a_n}$ , compared in Theorem 1.3.3.
binomial coefficient: the natural number $\binom{n}{k}$ given by Pascal’s rule, never by a quotient of factorials (Definition 1.2.1).
integer power: $a^{-n} \mathrel{:=} (a^{-1})^n$ for $a \ne 0$ in a field (Definition 1.5.1).
natural multiple: $n \cdot a$ , the sum of $n$ copies of $a$ , with its laws (Lemma 1.1.2).
$n$ -th root: the unique $b \ge 0$ with $b^n = a$ , written $\sqrt[n]{a}$ (Lemma 1.5.1).
rational power: $a^{m/n} \mathrel{:=} (\sqrt[n]{a})^m$ for $a > 0$ (Definition 1.5.2).
telescoping sum: a sum of terms $b_{k+1} - b_k$ , equal to the difference of its ends (Theorem 1.1.7).

Results used from earlier volumes

Nothing in the chapter is borrowed: everything is proved from the results below. No other chapter of this volume is used, since this chapter comes first.

Logic and Proof, Chapter 4:

induction (Logic and Proof, Axiom 4.2.1);
strengthening a claim before inducting (Logic and Proof, Example (Prove more to prove less)).

Sets and Functions, Chapter 3:

the recursion theorem (Sets and Functions, Theorem 3.3.1) and induction in practice (Sets and Functions, Theorem 3.2.1);
finite sums and products (Sets and Functions, Definition 3.3.1) and powers (Sets and Functions, Definition 3.3.2);
the laws of powers (Sets and Functions, Proposition 3.3.4);
the sum of the first $n$ numbers (Sets and Functions, Proposition 3.3.5) and the geometric sum (Sets and Functions, Proposition 3.3.6);
a finite list of reals has a largest term (Sets and Functions, Proposition 3.3.8);
Bernoulli’s inequality for real numbers (Sets and Functions, Theorem 3.3.7), used in Sections 1.2–1.5.

Sets and Functions, Chapter 5:

well-definedness, as a proof method (Sets and Functions, Example (Method: checking that a definition on classes is well defined)).

Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ :

its $\mathbb{N}$ and the $\mathbb{N}$ of Sets and Functions are the same (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Corollary 1.4.3), from its Chapter 1;
commutative rings and fields (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 2.3.1) and the algebra of a ring (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 2.3.2), from its Chapter 2;
the rules of an ordered ring, including the multiples of its Section 2.4 (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 2.4.3), and $\mathbb{Z}$ as an ordered ring (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 2.4.2);
reciprocals reverse the order (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 3.2.2), from its Chapter 3;
the rules of an ordered field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.1.2), the absolute value (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.1.3) and fractions in a field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Lemma 4.1.1), from its Chapter 4;
the natural numbers inside an ordered field (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 4.2.1, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.2.1);
the Archimedean property (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.2.3);
limits of sequences (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 4.3.2), the first limits (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Example (First Limits)), only the tail matters (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 4.3.2), and how to prove a limit from the definition (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Example (Method: proving a limit from the definition));
the completeness of $\mathbb{R}$ (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 5.5.2, Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Definition 5.5.1), from its Chapter 5;
a complete ordered field is Archimedean (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Theorem 6.1.2), from its Chapter 6;
square roots in $\mathbb{R}$ (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Proposition 6.1.4) and a point constructed as a supremum (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ , Exercise (B2: Cube roots)), from its Chapter 6, which Section 1.5 runs for every exponent.

Threads

From sums to Cauchy–Schwarz. The rules for finite sums of Section 1.1 are what Chapter 9 uses to prove the Cauchy–Schwarz inequality. Chapter 4 uses the same sums with terms in a field.
Binomial theorem and Fermat. The binomial coefficients are defined without division because Chapter 4 needs them in $\mathbb{F}_p$ , where $p!$ is zero. There $(x + y)^p = x^p + y^p$ .
Rational to real powers. Section 1.5 stops at rational exponents. Chapter 8 defines $a^x$ for every real $x$ from the exponential function. Analysis on the Real Line proves the facts about the exponential that Chapter 8 borrows.