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

Chapter 2 - The Real Number Line

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.

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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} \subseteq 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

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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 \subseteq 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 \subseteq 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|

Corollary. Given a, b \in \mathbb{R}, then ||a| - |b|| \leq |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. |x = r if and only if x = r or x = -r
2. |x < r if and only if -r < x < r
3. |x > r if either x > r or x < -r

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Definition. The standard distance function or metric on the real numbers \mathbb{R} given a, b is |a - b|.

Theorem. For any real numbers a, b, c,

1. |a - b| > 0 if and only if a \neq b and |a - b| = 0 if and only if a = b
2. |a - b| = |b - a|
3. |a - c| \leq |a - b| + |b + c|

Definition. A set together with a function satisfying these three properties is known as a metric space.

Definition. The $\varepsilon$-neighborhood of a \in \mathbb{R}, denoted V_\varepsilon(a) is the set of all real numbers x \in \mathbb{R} such that |x - a| < \varepsilon. That is,

V_\varepsilon(a) = (a - \varepsilon, a + \varepsilon)

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Decimals. Let x \in \mathbb{R} such that x > 0. By the archimedean property, there exists some b_0 \in \mathbb{N} \cup {0} such that b_0 < x < b_0 + 1. We can repeat this to see

x = b_0 + \frac{b_1}{10} + \frac{b_2}{100} + \ldots + \frac{b_n}{100^n} + \ldots

Definition. The decimal expansion of x is denoted b_0.b_1 b_2 b_3 \ldots.

Section 2.5 - Intervals

Definition. A subset I is an interval if and only if, given a, b \in I, then [a, b] \subseteq I.

Definition. Intervals I_1, I_2, \ldots, I_n, \ldots are nested if and only if I_1 \subseteq I_2 \subseteq \ldots \subseteq I_n \subseteq \ldots.

Theorem. Nested Intervals Property. If I_n = [a_n, b_n] is a set of nested intervals that are closed and bound, then there exists some number z \in \mathbb{R} such that z \in I_n for all n.

Theorem. If a < b, then the interval [a, b] is an uncountable set.

Corollary. \mathbb{R} is uncountable.