The Timpson Lyceum · Real Analysis (2020)

Lecture 2 (Real Analysis 2020) part 2

Overview

This part finishes the preliminary material on interval notation and then opens Section 1.1 with the first building block toward the completeness axiom: the notion of an inductive set. We use intervals to describe sets of real numbers compactly, taking care with the role of the infinity symbol. We then define what it means for a subset of $\mathbb{R}$ to be inductive, look at several examples, and use this idea to give a precise definition of the natural numbers as the intersection of all inductive subsets of $\mathbb{R}$ . This is the constructive, real-analysis way of pinning down $\mathbb{N}$ rather than assuming it.

Key Concepts

Interval notation. For real numbers $a \le b$ , intervals name all reals between two endpoints, with brackets recording whether each endpoint is included:

$(a,b)$ — all $x$ with $a < x < b$ (neither endpoint included).
$[a,b]$ — all $x$ with $a \le x \le b$ (both endpoints included).
$(a,b]$ — $a < x \le b$ ; $[a,b)$ — $a \le x < b$ (half-open).
$[a,a]$ — the single-element set $\{a\}$ (same number twice with closed brackets).

Infinity in intervals. $\infty$ is not a real number, so it is never an included endpoint — it always sits next to a parenthesis, and only signals "without end":

$[a,\infty)$ — all $x$ with $x \ge a$ ; $(a,\infty)$ — all $x$ with $x > a$ .
$(-\infty,b]$ — all $x$ with $x \le b$ ; $(-\infty,b)$ — all $x$ with $x < b$ .
$(-\infty,\infty)$ — all real numbers, $\mathbb{R}$ .

On a number line, $(a,\infty)$ is everything from $a$ rightward forever; $(-\infty,b)$ is everything from $b$ leftward forever.

Inductive set. A set $S \subseteq \mathbb{R}$ is inductive provided both:

$1 \in S$ , and
for every $x$ , if $x \in S$ then $x+1 \in S$ (closed under adding $1$ ).

Knowing only these two facts about a set forces a whole upward chain into it: $1 \in S \Rightarrow 2 \in S \Rightarrow 3 \in S \Rightarrow \cdots$ . If you are also told some other element is in $S$ — say $\tfrac12$ — then $\tfrac32, \tfrac52, \ldots$ must all be in $S$ as well.

Examples and non-examples.

$\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ are all inductive: each contains $1$ and is closed under $+1$ .
$[0,\infty)$ and $[1,\infty)$ are inductive.
The "half-step" set $\{\ldots, -\tfrac32, -1, -\tfrac12, 0, \tfrac12, 1, \tfrac32, \ldots\}$ is inductive — it contains $1$ , and any element plus $1$ is again in the set.
$\{1\}$ is not inductive: it contains $1$ but fails condition 2, since $1+1=2 \notin \{1\}$ .

Structural facts. An inductive set must extend upward without end — it can have no largest element, because any candidate largest element $k$ forces $k+1$ in too. It need not extend in the negative direction. Being a subset of $\mathbb{R}$ does not make a set inductive (every set here is a subset of $\mathbb{R}$ , but most are not inductive).

Natural numbers. The natural numbers $\mathbb{N}$ are defined as the intersection of every inductive subset of $\mathbb{R}$ : $\mathbb{N} = \bigcap \{\, S \subseteq \mathbb{R} : S \text{ is inductive} \,\}.$ The intersection keeps only the elements common to all inductive sets — exactly the numbers guaranteed present by the two conditions: $1, 2, 3, 4, \ldots$ . Note $0 \notin \mathbb{N}$ under this definition: $0$ fails to lie in every inductive set (e.g. $0 \notin [1,\infty)$ ), so it is excluded. (Whether $0$ is "natural" is a convention; defining $\mathbb{N}$ from $\mathbb{R}$ this way leaves it out.)

Worked Example

Claim. The half-open ray $[1,\infty)$ is an inductive set.

Verify the two defining conditions directly:

\begin{aligned} \textbf{(1)}\quad &1 \in [1,\infty) \quad\text{since } 1 \ge 1. \\ \textbf{(2)}\quad &\text{Suppose } x \in [1,\infty),\ \text{i.e. } x \ge 1. \\ &\text{Then } x + 1 \ge 1 + 1 = 2 \ge 1,\ \text{so } x+1 \in [1,\infty). \end{aligned}

Both conditions hold, so $[1,\infty)$ is inductive. By contrast $\{1\}$ fails condition (2): $1 \in \{1\}$ but $1+1 = 2 \notin \{1\}$ , so $\{1\}$ is not inductive — showing condition (1) alone is never enough.

Summary

Intervals name sets of reals by endpoints; brackets $[\ ]$ include an endpoint, parentheses $(\ )$ exclude it, and $[a,a]=\{a\}$ .
$\infty$ is not a real number, so it always pairs with a parenthesis and only means "continues without end."
A set $S \subseteq \mathbb{R}$ is inductive when $1 \in S$ and $x \in S \Rightarrow x+1 \in S$ .
An inductive set has no largest element and need not extend in the negative direction; $\mathbb{Z}$ , $\mathbb{Q}$ , $\mathbb{R}$ , and $[1,\infty)$ are inductive, while $\{1\}$ is not.
The natural numbers $\mathbb{N}$ are the intersection of all inductive subsets of $\mathbb{R}$ — the elements forced into every inductive set, namely $1, 2, 3, \ldots$
Under this construction $0$ is not a natural number, since it is absent from some inductive sets.