Serlo: EN: Properties of supremum and infimum

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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:

Now, for the supremum and infimum the following rules apply (where $A,B,D\subseteq ℝ$ and $f,g:D\to ℝ$ and $\lambda \in ℝ$). In the following it is always assumed that the supremum or the infimum exists.

Rules for the supremum

• $\sup A\ge \inf A$
• $A\subseteq B\Rightarrow \sup A\le \sup B$

• $\sup (A\cup B)=\max \{\sup A,\sup B\}$
• $\sup (A\cap B)\le \min \{\sup A,\sup B\}$
• $\sup (-A)=\sup (\{-x:x\in A\})=-\inf (A)$
• $\sup (\{x^{-1}:x\in A\})=(\inf (A){)}^{-1}$, if $\inf (A)>0$ holds.
• $\sup (\lambda A)=\sup (\{\lambda x:x\in A\})=\lambda \cdot \sup (A)$ for $\lambda \ge 0$
• $\sup (A+B)=\sup (\{x+y:x\in A\wedge y\in B\})=\sup (A)+\sup (B)$
• $\sup (A\cdot B)=\sup (\{x\cdot y:x\in A\wedge y\in B\})=\sup (A)\cdot \sup (B)$, if $A$ and $B$ contain only non-negative elements.
• $\sup (f+g)(D)\le \sup f(D)+\sup g(D)$
• There is a sequence $(a_n{)}_{n\in \mathbb{N}}$ in $A$ with $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=\sup A$.

The supremum of a sum of two functions might be smaller than the sum of its suprema.

Rules for the infimum

• $\inf A\le \sup A$
• $A\subseteq B\Rightarrow \inf A\ge \inf B$
• $\inf (A\cup B)=\min \{\inf A,\inf B\}$
• $\inf (A\cap B)\ge \max \{\inf A,\inf B\}$
• $\inf (-A)=\inf (\{-x:x\in A\})=-\sup (A)$
• $\inf (\lambda A)=\inf (\{\lambda x:x\in A\})=\lambda \cdot \inf (A)$ for $\lambda \ge 0$
• $\inf (A+B)=\inf (\{x+y:x\in A\wedge y\in B\})=\inf (A)+\inf (B)$
• $\inf (A\cdot B)=\inf (\{x\cdot y:x\in A\wedge y\in B\})=\inf (A)\cdot \inf (B)$, if $A$ and $B$ contain only non-negative elements.
• $\inf (f+g)(D)\ge \inf f(D)+\inf g(D)$
• There is a sequence $(a_n{)}_{n\in \mathbb{N}}$ in $A$ with $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=\inf A$.

Proof of the rules

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

Existence of a sequence $(a_n{)}_{n\in \mathbb{N}}$ in $A$ with $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=\sup A$

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