02 Real Numbers, Min, Max, Bounds, Supremum, Infimum, Cardinality, Euclidian Space

Lecture from: 21.02.2024 | Video: Video ETHZ

Real Numbers

This section delves deeper into the properties of real numbers, introduces the concept of infinity, defines intervals and explores the crucial concepts of bounds, suprema, and infima.

Defining Minimum, Maximum, and Absolute Value

For $x,y\in \mathbb{R}$:

1. Maximum: $\max \{x,y\}:=\{\begin{array}{ll}x& \text{if }x\ge y\\ y& \text{if }y>x\end{array}$
2. Minimum: $\min \{x,y\}:=\{\begin{array}{ll}x& \text{if }x\le y\\ y& \text{if }y<x\end{array}$
3. Absolute Value: $|x|:=\max \{x,-x\}$ represents the distance of $x$ from 0.

Theorems on Absolute Value

Theorem: For all $x,y\in \mathbb{R}$:

1. Non-negativity: $|x|\ge 0$
2. Triangle Inequality: $|x+y|\le |x|+|y|$
3. Reverse Triangle Inequality: $|x-y|\ge ||x|-|y||$

Proofs of Absolute Value Theorems

We’ll use case distinction for these proofs.

Proof of Theorem 1 (Non-negativity)

• Case 1: $x\ge 0$: Then $|x|=\max \{x,-x\}=x\ge 0$.
• Case 2: $x<0$: Then $-x>0$, and $|x|=\max \{x,-x\}=-x>0$.

Therefore, in both cases, $|x|\ge 0$.

Proof of Theorem 2 (Triangle Inequality)

• Case 1: $x\ge 0,y\ge 0$: Then $x+y\ge 0$, so $|x+y|=x+y=|x|+|y|$.
• Case 2: $x<0,y<0$: Then $x+y<0$, so $|x+y|=-(x+y)=(-x)+(-y)=|x|+|y|$.
• Case 3: $x\ge 0,y<0$:
  • Subcase a) $x+y\ge 0$: $|x+y|=x+y\le x+(-y)=|x|+|y|$, since $y<0<-y$.
  • Subcase b) $x+y<0$: $|x+y|=-(x+y)=(-x)+(-y)$. Since $x\ge 0$, $-x\le 0\le x=|x|$, and since $y<0$, $-y=|y|$. Thus we have $|x+y|\le |x|+|y|$.
• Case 4: $x<0,y\ge 0$: Analogous to Case 3.

In all cases, $|x+y|\le |x|+|y|$.

Infinity

We introduce the symbols $\infty$ (infinity) and $-\infty$ (negative infinity) with the convention that $-\infty <x<\infty$ for all $x\in \mathbb{R}$. Note:

Infinity is not a real number:

$\infty$ and $-\infty$ are not elements of $\mathbb{R}$. The Archimedean Principle implies that for any real number, we can always find a larger (or smaller) natural number. If infinity were a number then it would violate the Archimedean Principle.

Intervals

Definition: An interval is a subset of $\mathbb{R}$ such that if it contains two distinct numbers, it also contains all real numbers between them. Notation:

1. Closed Interval: $[a,b]=\{x\in \mathbb{R}\mid a\le x\le b\}$ (includes endpoints). Can also be written as [a,b].
2. Open Interval: $(a,b)=\{x\in \mathbb{R}\mid a<x<b\}$ (excludes endpoints). Can also be written as ]a,b[.
3. Half-Open Intervals: $[a,b)=\{x\in \mathbb{R}\mid a\le x<b\}$ and $(a,b]=\{x\in \mathbb{R}\mid a<x\le b\}$ (includes one endpoint). Can also be written as [a,b[ or ]a,b] respectively.

Infinite Intervals: Let $a\in \mathbb{R}$. We can also define intervals extending to infinity:

• $(a,\infty )=\{x\in \mathbb{R}\mid x>a\}$
• $[a,\infty )=\{x\in \mathbb{R}\mid x\ge a\}$
• $(-\infty ,a)=\{x\in \mathbb{R}\mid x<a\}$
• $(-\infty ,a]=\{x\in \mathbb{R}\mid x\le a\}$
• $(-\infty ,\infty )=\mathbb{R}$

Bounds, Supremum, and Infimum

Let $A\subseteq \mathbb{R}$.

Definitions

• Upper Bound: $c\in \mathbb{R}$ is an upper bound of $A$ if for all $a\in A$, $a\le c$. $A$ is bounded above if it has an upper bound.
• Lower Bound: $c\in \mathbb{R}$ is a lower bound of $A$ if for all $a\in A$, $a\ge c$. $A$ is bounded below if it has a lower bound.
• Maximum: If $A$ is bounded above and has an upper bound that belongs to the set $A$ itself, that element is called the maximum of $A$.
• Minimum: If $A$ is bounded below and has a lower bound that belongs to $A$, that element is called the minimum of $A$.
• Bounded: A set $A$ is bounded if it is bounded both above and below.

Examples:

• The interval $[0,1]$ is bounded; its maximum is 1, and its minimum is 0.
• The interval $[0,1)$ is bounded; its minimum is 0, but it has no maximum. 1 is an upper bound, but $1\notin [0,1)$. However, any number larger than 1 is also an upper bound.

Showing that a number is not upper/lower bound

To prove a number $c$ is not an upper (or lower) bound, find an element $a\in A$ such that $a>c$ (or $a<c$ for a lower bound). Often, you can construct such an element using the average of $c$ and a known upper bound (or lower bound).

Supremum and Infimum

Definition of Supremum and Infimum

• Supremum (Least Upper Bound): If a set $A\subseteq \mathbb{R}$ is bounded above, the supremum of $A$, denoted $\sup (A)$, is the least upper bound of $A$. It is the smallest number that is greater than or equal to every element in $A$.

• Infimum (Greatest Lower Bound): If a set $A\subseteq \mathbb{R}$ is bounded below, the infimum of $A$, denoted $\inf (A)$, is the greatest lower bound of $A$. It is the largest number that is less than or equal to every element in $A$.

Relationship with Maximum and Minimum

• If a set $A$ has a maximum element, then $\sup (A)=\max (A)$. The supremum coincides with the maximum when the maximum exists.
• If a set $A$ has a minimum element, then $\inf (A)=\min (A)$. The infimum coincides with the minimum when the minimum exists.
• A set may have a supremum (or infimum) without having a maximum (or minimum). This typically occurs when a set is bounded but doesn’t contain its least upper bound (or greatest lower bound).

The Empty Set Convention

If $A=\varnothing$ (the empty set), then, by convention:

• $\sup (\varnothing )=-\infty$
• $\inf (\varnothing )=\infty$

The why…

This convention arises from the vacuous truth that every real number is both an upper and a lower bound for the empty set (as there are no elements to contradict this). Setting the supremum to $-\infty$ and the infimum to $\infty$ maintains consistency in certain theorems and definitions.

Illustrative Examples

Example 1: A Half-Open Interval

Let $A=[1,2)$.

• $\inf (A)=1$
• $\sup (A)=2$https://video.ethz.ch/lectures/d-math/2024/spring/401-0212-16L/f18cbc47-2906-4d5c-bfd5-dd8ae5cd5bc8.html
• $\min (A)=1$ (The minimum exists and is equal to the infimum).
• $\max (A)$ does not exist (The set does not contain its least upper bound). 2 is not an element of A.

Example 2: The Harmonic Series (Partial Sums)

Let $A=\{1,1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3},\dots \}$. This set represents the partial sums of the harmonic series.

• $\min (A)=1$ (The smallest element is the first term).
• $\inf (A)=1$ (The greatest lower bound is also 1).
• $\max (A)$ does not exist.

Proof that the maximum does not exist: The harmonic series is known to diverge (it does not converge to a finite limit). This can be demonstrated using a simple comparison test:

Since we can always group consecutive terms to obtain sums greater than or equal to $\frac{1}{2}$, the sum of the harmonic series grows without bound (tends to infinity). Therefore, there is no largest element (maximum) in the set of partial sums.

As a consequence, $\sup (A)$ does not exist.

Theorem:

Let $A\subseteq \mathbb{R}$ be a non-empty set that is bounded above, and let $M$ be the set of all upper bounds of $A$. Then $\inf (M)$ exists and $\inf (M)=\sup (A)$.

Proof

1. $M$ is non-empty and bounded below: Since $A$ is bounded above, there exists at least one upper bound, so $M\ne \varnothing$. Furthermore, every element of $A$ is a lower bound for $M$ (by the definition of an upper bound). Since $A$ is non-empty, $M$ is bounded below.

2. Existence of $\inf (M)$: Because $M$ is non-empty and bounded below, by the completeness axiom, $\inf (M)$ exists. Let $s=\inf (M)$.

3. Show $s$ is an upper bound for $A$ ($s\in M$): Suppose, for contradiction, that $s$ is not an upper bound for $A$. Then there exists an $a\in A$ such that $a>s$. Since $s$ is the greatest lower bound of $M$, any number less than $s$ cannot be a lower bound. Thus, if we consider the average of $s$ and $a$ which lies between them ( $\frac{s+a}{2}$ ), this average must be in $M$ (i.e an upper bound to $A$). This is a contradiction since it would be an upper bound that is strictly less than the greatest lower bound of $M$. Therefore, $s$ must be an upper bound for $A$, so $s\in M$.

4. Show $s$ is the least upper bound for $A$ ($s=\sup (A)$): Since $s=\inf (M)$, $s$ is a lower bound for $M$. This means that for any $m\in M$ (i.e., any upper bound of $A$), we have $s\le m$. This shows that $s$ is less than or equal to any upper bound of $A$.

   Combined with the fact that $s$ is itself an upper bound of $A$ (from step 3), this means $s$ is the least upper bound of $A$. Therefore, $s=\sup (A)$.

5. Conclusion: We have shown that $\inf (M)=s=\sup (A)$. Therefore, $\inf (M)=\sup (A)$.

This theorem establishes a fundamental relationship between the supremum of a set and the infimum of its set of upper bounds.

Cardinality: Measuring the Size of Sets

Cardinality provides a way to compare the sizes of sets, even infinite ones. We use the following terminology:

Injective, Surjective, and Bijective Functions

These function types are fundamental for comparing set sizes:

• Injective (One-to-one): A function $f:A\to B$ is injective if each element in $B$ is mapped to by at most one element in $A$. Formally: $f(x_1)=f(x_2)⟹x_1=x_2$.
• Surjective (Onto): A function $f:A\to B$ is surjective if every element in $B$ is mapped to by at least one element in $A$. Formally: $\forall b\in B,\exists a\in A:f(a)=b$.
• Bijective (One-to-one and Onto): A function is bijective if it is both injective and surjective. A bijection establishes a one-to-one correspondence between the elements of two sets.

Equinumerosity (Equimächtig)

Two sets $A$ and $B$ are said to be equinumerous (or have the same cardinality) if there exists a bijective function $f:A\to B$. This is denoted as $|A|=|B|$. Intuitively, this means the elements of $A$ and $B$ can be paired up perfectly.

Classifying Cardinalities

• Finite Sets: A set is finite if it is equinumerous to the set $\{1,2,\dots ,n\}$ for some $n\in \mathbb{N}$. Its cardinality is $n$.
• Countably Infinite Sets: A set is countably infinite if it is equinumerous to the set of natural numbers $\mathbb{N}$. Examples include the integers ($\mathbb{Z}$) and the rational numbers ($\mathbb{Q}$).
• Countable Sets: A set is countable if it is finite or countably infinite.
• Uncountably Infinite Sets: A set is uncountably infinite if it is infinite and not countably infinite. The real numbers ($\mathbb{R}$) are an example of an uncountably infinite set. Cantor’s diagonal argument famously proves this.

Euclidean Space: ${\mathbb{R}}^n$

Definition

For $n\ge 1$, the Euclidean space ${\mathbb{R}}^n$ is the set of all ordered n-tuples of real numbers:

${\mathbb{R}}^n=\{(x_1,x_2,\dots ,x_n)\mid x_i\in \mathbb{R}\text{ for }i=1,2,\dots ,n\}$

Operations in ${\mathbb{R}}^n$

${\mathbb{R}}^n$ forms a vector space over the real numbers with the following operations:

• Vector Addition: For $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)$ in ${\mathbb{R}}^n$, $x+y=(x_1+y_1,\dots ,x_n+y_n)$.
• Scalar Multiplication: For $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ and $\alpha \in \mathbb{R}$, $\alpha \cdot x=(\alpha x_1,\dots ,\alpha x_n)$.

Scalar Product (Dot Product)

The scalar product (or dot product) in ${\mathbb{R}}^n$ is defined as: $x\cdot y={\sum }_{i=1}^n{x}_i{y}_i=x_1{y}_1+x_2{y}_2+...+x_n{y}_n$ Properties of the Scalar Product:

• Symmetry: $x\cdot y=y\cdot x$
• Bilinearity: $(\alpha x+\beta y)\cdot z=\alpha (x\cdot z)+\beta (y\cdot z)$ (and similarly in the second argument).
• Positive Definiteness: $x\cdot x\ge 0$, and $x\cdot x=0$ if and only if $x=0$ (the zero vector).

Orthogonality

Two vectors $x$ and $y$ in ${\mathbb{R}}^n$ are orthogonal if their scalar product is zero: $x\cdot y=0$.

Norms

A norm on ${\mathbb{R}}^n$ is a function $||\cdot ||:{\mathbb{R}}^n\to \mathbb{R}$ that satisfies: The most common norm is the Euclidean norm (or 2-norm):

$||x||=||x|{|}_2=\sqrt{x\cdot x}=\sqrt{{\sum }_{i=1}^n{x}_i^2}$

This represents the geometric length of the vector $x$. Other norms include the 1-norm ($||x|{|}_1={\sum }_{i=1}^n|x_i|$) and the infinity-norm ($||x|{|}_{\infty }={\max }_{1\le i\le n}|x_i|$).

Cauchy-Schwarz Inequality

Theorem (Cauchy-Schwarz Inequality): For all $x,y\in {\mathbb{R}}^n$, $|x\cdot y|\le ||x||\cdot ||y||$ with equality if and only if $x$ and $y$ are linearly dependent.

Properties of the Euclidean Norm

Theorem: For all $x\in {\mathbb{R}}^n$ and $\alpha \in \mathbb{R}$:

1. Non-negativity: $||x||\ge 0$
2. Absolute Homogeneity: $||\alpha \cdot x||=|\alpha |\cdot ||x||$
3. Triangle Inequality: $||x+y||\le ||x||+||y||$

Proof of the Triangle Inequality using Cauchy-Schwarz:

To prove $||x+y||\le ||x||+||y||$, we will work with the squares of the norms, which are non-negative:

Now, since both $||x+y||$ and $||x||+||y||$ are non-negative, we can take the square root of both sides of the inequality $||x+y|{|}^2\le (||x||+||y||{)}^2$ without changing the direction of the inequality:

$\sqrt{||x+y|{|}^2}\le \sqrt{(||x||+||y||{)}^2}$

This gives us the desired Triangle Inequality:

$||x+y||\le ||x||+||y||$

Continue here: 03 Supremum, Infimum, Cross Product, Complex Numbers, Sequences