2.5 Intervals

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• TODO
  • Proofs

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Definition: Bounded interval

Let $a,b\in \mathbb{R}$ and $a\le b$.

• Open interval: $(a,b):=\{x\in \mathbb{R}:a<x<b\}$.
• Closed interval: $[a,b]:=\{x\in \mathbb{R}:a\le x\le b\}=(a,b)\cup \{a,b\}$.
• Half open/closed interval:
  • $[a,b):=(a,b)\cup \{a\}$ and
  • $(a,b]:=(a,b)\cup \{b\}$.

NOTE

$a,b$: Endpoints of an interval

$b-a$: Length of an interval

• $(a,a)=\varnothing$
• $[a,a]=a$

Definition: Unbounded interval

Let $a,b\in \mathbb{R}$

• Infinite open intervals: $(a,\infty ):=\{x\in \mathbb{R}:x>a\}$ and $(-\infty ,b):=\{x\in \mathbb{R}:x<b\}$.
• Infinite closed intervals: $[a,\infty ):=\{x\in \mathbb{R}:x\ge a\}$ and $(-\infty ,b]:=\{x\in \mathbb{R}:x\le b\}$.
• $(-\infty ,\infty ):=\mathbb{R}$.

2.5.1 Theorem: Properties of intervals

If $S$ is a subset of $\mathbb{R}$ that contains at least two points and has the property

$$x,y\in S\wedge x<y\to [x,y]\subseteq S,$$

then $S$ is an interval.

2.5.2 Theorem: Properties of nested interals

If $I_n=[a_n,b_n],n\in \mathbb{N}$ is a nested sequence of closed bounded intervals, then there exists a number $\xi \in \mathbb{R}$ such that $\xi \in I_n$ for all $n\in \mathbb{N}$.

2.5.3 Theorem: Convergence of shrinking intervals

If $I_n:=[a_n,b_n],n\in \mathbb{N}$, is a nested sequence of closed, bounded intervals such that the lengths $b_n-a_n$ of $I_n$ satistfy

$$\inf \{b_n-a_n:n\in \mathbb{N}\}=0,$$

then the number $\xi$ contained in $I_n$ for all $n\in \mathbb{N}$ is unique.

2.5.4 Theorem: Cantor’s first proof

The set of real numbers $\mathbb{R}$ is not countable.

2.5.5 Theorem: Cantor’s second proof

The unit interval $[0,1]:=\{x\in \mathbb{R}:0\le x\le 1\}$ is not countable.