notes-archive/docs/math/real-analysis/2-reals.md

Chapter 2

Section 2.1 - The Algebraic and Order Properties of Real Numbers

Proposition. 2.1.1: \mathbb{R} is a field, with zero element 0 and identity 1.

Definition. The rational numbers \mathbb{Q} is the field of fractions of the natural numbers \mathbb{N}.

Theorem. 2.1.4: There does not exist a rational number r such that r^2 = 2.

---

Definition. An ordered field is a field F together with subset F^+ such that

1. F+ is closed under addition and multiplication
2. If a \in F, then exclusively a \in F^+, a = 0, or -a \in F^+.

Theorem. In any ordered field F, the following hold

1. 1 \in F^+
2. \mathbb{N} \subset F^+
3. If a \in F^+, then \frac{1}{a} \in F^+

Definition The order relation a > b and b < a is defined by a - b \in F^+.

Theorem. If a, b, c \in F, then

1. One of a > b, a = b, or a < b hold (trichotomy)
2. If a > b and b > c, then a > c (transitivity)
3. If a > b, then -a < -b
4. If a > b and c > 0, then ac > bc
5. If a > b and c < 0, then ac < bc
6. If a > b > 0, then \frac{1}{b} > \frac{1}{a} > 0

---

Definition. Let S be a nonempty subset of ordered field F. Then, S is bounded above if there exists some u \in F such that s \leq u for all s \in S. Then, said element u is an upper bound of S.

Definition. Let S be a nonempty subset of ordered field F. Then, S is bounded below if there exists some u \in F such that s \geq u for all s \in S. Then, said element u is a lower bound of S.

Definition. Let S be a nonempty subset of ordered field F. Then, S is bounded if it is bounded both above and below.

Definition. Given field F and nonempty subset S \subset F, an element u \in F is a supremum or least upper bound of S if u is an upper bound of S, and given any other upper bound v, then u < v

Definition. Given field F and nonempty subset S \subset F, an element u \in F is an infimum or greatest lower bound of S if u is a lower bound of S, and given any other lower bound v, then u > v

Definition. Given an ordered field F, the field has the supremum/infimum property if given any nonempty subset S, if S is bounded above/below, S has a supremum/infimum.

Section 2.2 - Absolute Value and the Real Line

Definition. Absolute value is defined as normal (piecewise). Multiline function in LaTeX are hard.

Theorem. Given any a, b \in \mathbb{R}, we know that

1. |a| > 0 for a \neq 0
2. |ab| = |a||b|
3. |a + b| \leq |a| + |b|

Collary. Givem a, b \in \mathbb{R}, then \abs{\abs{a} - \abs{b}} \leq \abs{a - b}.

Remark. Every field has at least one absolute value function.

Theorem. In an ordered field F, for any r > 0, we know that

1. \abs x = r if and only if x = r or x = -r
2. \abs x < r if and only if -r < x < r
3. \abs x > r if either x > r or x < -r