Serlo: EN: How to prove existence of a supremum or infimum

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}}

General procedure

How do we find the supremum/infimum of a set? there are some strategies:

1. Visualize the set: What does the set look like? Try to draw it. You may use a paper, a computer or your imagination!
2. Make a hypotheses: Is the set bounded from above? if yes, what is a suitable candidate for a supremum? And why? If not, why is it unbounded? And what about the supremum?
3. Find a proof for the supremum/ infimum: Now, how can you show that the suspected supremum/ infimum is indeed one? Or that there is no supremum or infimum due to unboundedness? Take a piece of paper and try to construct a proof. Perhaps, within the proof you may notice that your first step was wrong for some reason. This is a good sign: detecting your own wrong thoughts takes you closer to the actual truth.
4. Write down the proof: If you have a plan how to make a proof, formulate it in a short and understandable way - such that anyone who reads it and doesn't know the solution gets the answer as quickly and nicely as possible.

General structure of proof

Proofs that a set has a supremum/ infimum generally follow some patterns, which we want to list here:

Supremum: structure of proof

In order to show that a number $s$ is the supremum of a set $M$ , you may proceed as follows:

1. Prove that $s$ is an upper bound of $M$ : This is done showing $y\le s$ for all $y\in M$.
2. Prove that no number $x<s$ is an upper bound of $M$: Take any $x<s$ and show that there is a $y\in M$ with $y>x$.

Infimum: structure of proof

A proof that $\tilde{s}$ is the infimum of a set $M$ , could look as follows:

1. Prove that $\tilde{s}$ is a lower bound of $M$ : This is done showing $y\ge \tilde{s}$ for all $y\in M$.
2. Prove that no number $x>\tilde{s}$ is a lower bound of $M$ : Take any $x>\tilde{s}$ and show that there is a $y\in M$ with $y<x$.

Maximum: structure of proof

the proof makes use of the definition of the maximum:

1. Prove that $m$ is an upper bound of $M$: This is done showing $y\le m$ for all $y\in M$.
2. Prove that $m\in M$.

Minimum: structure of proof

In order to show that $\tilde{m}$ is the minimum of $M$ , you may proceed as for the maximum:

1. Prove that $\tilde{m}$ is a lower bound of $M$: This is done showing $y\ge \tilde{m}$ for all $y\in M$.
2. Prove that $\tilde{m}\in M$.

Examples for determining supremum/infimum

Finite sets

With finite sets of real numbers, determining the infimum and supremum is simple. These sets must always have a maximum (greatest number) and a minimum (smallest number). The maximum of the set is automatically supremum and the minimum is automatically infimum of the set.

Intervals

The determination of the infimum and supremum for intervals is quite simple, because the lower boundary point is always the infimum and the upper boundary point is always the supremum:

Sequences

The set $M=\{5+\frac{2}{n}:n\in \mathbb{N}\}$.

Suprema and infima of sequence elements are often required in mathematics. As those elements form a set, the supremum and infimum is well-defined. The set will have often infinitely, but always countably many elements. Let's start with an example:

Sets of function values

{{#invoke:Mathe für Nicht-Freaks/Seite|unten}}