# Math for Non-Geeks/Properties of suprema and infima

Since the Supremum is applied to sets, a very obvious question is: What happens if we change the set? If we intersect it with another set, for example, or take the union with another set, or if we make it larger or smaller? Here we will learn some rules that will help you to work with the Supremum in such cases.

## Overview about rules for supremum and infimum

First, let us introduce the following abbreviations:
Math for Non-Geeks: Template:Box
Now, for the supremum and infimum the following rules apply (where $A,B,D\subseteq \mathbb {R}$ and $f,g:D\rightarrow \mathbb {R}$ and $\lambda \in \mathbb {R}$). In the following it is always assumed that the supremum or the infimum exists.

### Rules for the supremum

* $\sup A\geq \inf A$
* $A\subseteq B\Rightarrow \sup A\leq \sup B$
* $\sup(A\cup B)=\max\{\sup A,\sup B\}$
* $\sup(A\cap B)\leq \min\{\sup A,\sup B\}$
* $\sup(-A)=\sup(\{-x:x\in A\})=-\inf(A)$
* $\sup(\{x^{-1}:x\in A\})=(\inf(A))^{-1}$, if $\inf(A)>0$ holds.
* $\sup(\lambda A)=\sup(\{\lambda x:x\in A\})=\lambda \cdot \sup(A)$ for $\lambda \geq 0$
* $\sup(A+B)=\sup(\{x+y:x\in A\land y\in B\})=\sup(A)+\sup(B)$
* $\sup(A\cdot B)=\sup(\{x\cdot y:x\in A\land y\in B\})=\sup(A)\cdot \sup(B)$, if $A$ and $B$ contain only non-negative elements.
* $\sup(f+g)(D)\leq \sup f(D)+\sup g(D)$
* There is a sequence $(a_{n})_{n\in \mathbb {N} }$ in $A$ with $\lim _{n\rightarrow \infty }a_{n}=\sup A$.
Question: Why does $\sup A\leq \sup B\Rightarrow A\subseteq B$ not hold? Find a counterexample!
A counterexample is $A:=\{1\},B:=\{2\}$.
Question: Why does $\sup(A\cap B)=\min\{\sup A,\sup B\}$ not hold? Find a counterexample!
Let $A=\{1,2\}$ and $B=\{1,3\}$. Then $A\cap B=\{1\}$ and hence $\sup A\cap B=1$, but $\sup A=2$ and $\sup B=3$, so $\min\{\sup A,\sup B\}=2$.
Question: Why does $\sup(f+g)(D)=\sup f(D)+\sup g(D)$ not hold? Find a counterexample!
We set $D:=[0,1]$. As a function, we choose $f:D\to \mathbb {R} ,x\mapsto x$ and $g:D\to \mathbb {R} ,x\mapsto 1-x$. Then $(f+g):D\to \mathbb {R} ,x\mapsto 1$, so $\sup(f+g)(D)=1<1+1=\sup f(D)+\sup g(D)$

### Rules for the infimum

* $\inf A\leq \sup A$
* $A\subseteq B\Rightarrow \inf A\geq \inf B$
* $\inf(A\cup B)=\min\{\inf A,\inf B\}$
* $\inf(A\cap B)\geq \max\{\inf A,\inf B\}$
* $\inf(-A)=\inf(\{-x:x\in A\})=-\sup(A)$
* $\inf(\lambda A)=\inf(\{\lambda x:x\in A\})=\lambda \cdot \inf(A)$ for $\lambda \geq 0$
* $\inf(A+B)=\inf(\{x+y:x\in A\land y\in B\})=\inf(A)+\inf(B)$
* $\inf(A\cdot B)=\inf(\{x\cdot y:x\in A\land y\in B\})=\inf(A)\cdot \inf(B)$, if $A$ and $B$ contain only non-negative elements.
* $\inf(f+g)(D)\geq \inf f(D)+\inf g(D)$
* There is a sequence $(a_{n})_{n\in \mathbb {N} }$ in $A$ with $\lim _{n\rightarrow \infty }a_{n}=\inf A$.

## Proof of the rules

In the following sections we will prove the above properties only for the Supremum. The infimum is treated analogously.

### Supremum is greater/equal to the infimum

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Estimating the supremum of subsets

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum and union

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum and intersection

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum and multiplication with $-1$

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis
Math for Non-Geeks: Template:Box

### Supremum and multiplication with a (non-negative) number

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum and sums

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum and products

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Supremum of the sum is smaller/equal the sum of suprema

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis

### Existence of a sequence $(a_{n})_{n\in \mathbb {N} }$ in $A$ with $\lim _{n\to \infty }a_{n}=\sup A$

Math for Non-Geeks: Template:BoxMath for Non-Geeks: Template:Beweis