Lecture 1

What are real numbers?

There are two methods to construct real numbers:

• Dedekind (1831-1916)
  Dedekind cut: If $x \in A^-, y \in A^+ \Rightarrow x \lt y,$ then the cut $(A^-, A^+)$ determines a real number. But elementary arithmetic is cumbersome; think about $(A^-, A^+) + (B^-, B^+).$

• Cantor (1845-1918)
  Define real numbers with sequences, which makes elementary arithmetic easier. If $X = \{ x_n \in \mathbb{Q} \}$ and $Y = \{ y_n \in \mathbb{Q} \}$ converge to the same number, say, $\sqrt{2},$ then they should be “equal”.

Def. $X \leq Y$ if each upper bound of $Y$ is also an upper bound of $X.$
Remark:

1. Reflexive: $X \leq X.$
2. Transitive: $X \leq Y$ and $Y \leq Z \Rightarrow X \leq Z.$

Let $S$ be the set of all bounded, non-decreasing, and rational sequences. Then $\langle S, \leq \rangle$ is an ordered set.

Let $X \sim Y$ denote $X \geq Y$ and $X \leq Y.$ Then $S$ can be split into equivalent classes. The set of all such equivalent classes is called the real numbers.

Exercise:

• Prove the equivalent classes defined by $\sim$ have good properties:
  1. $X \sim X$
  2. $X \sim Y \Rightarrow Y \sim X$
  3. $X \sim Y$ and $Y \sim Z \Rightarrow X \sim Z$

• Prove elementary arithmetic has good properties:
  • Existence of the $\mathbf{0}$ element
  • Commutativity of addition and multiplication
  • Associativity of addition and multiplication
  • etc.

• Prove law of trichotomy

Def. (Addition)
Let $\xi, \eta \in \mathbb{R},$ that is, $\xi \sim X = \{ x_1, \dotso \}$ and $\eta \sim Y = \{ y_1, \dotso \}$ for some sequence $X$ and $Y.$ Define $\xi + \eta \stackrel{\small{\text{def}}}{\sim} \{ x_1 + y_1, \dotso \}.$

Remark:

1. The $\mathbf{0} = \{ 0, \dotso \}$ element exists
2. Commutativity: $\xi + \eta = \eta + \xi$
3. Associativity: $(\xi + \eta) + \zeta = \xi + (\eta + \zeta)$
4. Existence of additive inverse: $\forall \xi \exists \eta, \xi + \eta = 0$ (Note: $Y = \{ -x_n \}$ is not non-decreasing.)
5. Total order; thus $\xi \gt \eta \Rightarrow \xi + \zeta \gt \eta + \zeta.$ Note: If $\xi = \{ x_n \} \lt \eta = \{ y_n \},$ then $\exists r \in \mathbb{Q}$ such that $r$ is an upper bound of $\xi$ but not an upper bound of $\eta.$ That is, $\exists N$ such that $r \lt y_n$ for all $n \gt N.$ Thus $\exists N$ such that $x_k \lt y_n$ for all $k$ and for all $n \gt N.$

Proof. (Existence of additive inverse)
Idea: Find a non-increasing sequence $\{ y_n \} \to \xi,$ then $\{ -y_n \}$ is non-decreasing and is the additive inverse of $\{ x_n \}.$
Let $r \in \mathbb{Q}$ be an upper bound $\{ x_n \}.$
Make $y_1 = r,$ and use bisection method to compute $y_2.$ Let $m = \frac{x_1 + y_1}{2}:$

1. If $m$ is an upper bound of $\xi,$ then make $y_2 = m.$
2. Otherwise make $y_2 = y_1.$ Pick some $x_i \gt m$ and apply bisection method on $(x_i, y_2).$