notes.md

Understanding Analysis

Section 1.2: Some Preliminaries

Theorem 1.1.1

There is not rational number whose square is 2

Definition of Rational Number

A number that can be expressed in the form of p/q, where p and q are integers

Natural Numbers Set

N = {1, 2, 3, ...}

Set

A set is a collection of objects, the objects being the elements of the set Similar to a JS object, these objects can be a wide range of things like numbers to functions

De Morgan's Law

$(A \cap B)^c = A^c \cup B^c$ and $(A \cup B)^c = A^c \cap B^c

functions

a function is a mapping given a set A (domain) that associates each element of A to a single element from the set B (range) We can write it as $f: A -> B$ ${y \in B: y = f(x) for some x \in A}

Absolute Function

shown as |x|, it is a function that will convert any input number into the positive form, that is: if x < 0, then -x if x >= 0, then x This can be shown as the magnitude of the number and

Triangle Inequality

(i) |ab| = |a||b| (ii) |a + b| = |a| + |b|

Theorem 1.2.6

2 real numbers a and b are equal if for every e > 0, |a - b| < e

Section 1.3: Axiom of Completeness

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound

Definition 1.3.1

a set $A \subseteq R$, is bounded above if there exists a $b \in R$ such that b > a for all $a \in A$ Vice versa if it's bounded below

Definition 1.3.2

a real number is is the last upper bound for a set $A \subseteq R$ if: (i) s is an upper bound of A (ii) if b is any upper bound of A, then $b &gt;= s$ This is called the supremum of A, also notated as $s = Sup A$ The vice versa would be called the infimum of A

Definition 1.3.3

the element $a_0$ is the maximum of A if $a_0 &gt;= a$ for every $a \in A$ and $a_0 \in A$ similarly, the minimum of A is the vice versa

Lemma 1.3.8

Assume s is an upper bound for the set A then $s = sup A$ if and only if for every choice e > 0 there exists an element $a \in A$ such that s - e < a

Section 1.4: Consequences of Completeness

Theorem 1.4.1 (Nested Interval Property)

for each $n \in N$ assume we are given a closed interval $I_n = [a_n, b_n] = {x \in R: a_n &lt;= x &lt;= b_n}$ Assume also that each $I_n contains I_{n+1}$, so:

$I_1 \supseteq I_2 \supseteq I_3 \supseteq ...$

Then the intersect of these closed intervals is not the empty set, $\cap_{n=1}^{\infty} I_n \neq \emptyset$

Theorem 1.4.2 (Archimedean Property)

(i) Given any number $x \in R$ there exists an $n \in N$ satisfying n > x (ii) Given any real number y > 0 there exists an $n \in N$ satisfying $1/n &lt; y$

Theorem 1.4.3 (Density of Q in R)

for every 2 real numbers, a and b such that a < b, there exists a rational number r satisfying a < r < b

Corollary 1.4.4

Given any 2 real numbers a and b such that a < b, then there exists an irrational number t satisfying a < t < b

Theorem 1.4.5

There exists a real number $a \in R$ satisfying $a^2 = 2$