How to prove existence of a supremum or infimum – Serlo

General procedureBearbeiten

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 proofBearbeiten

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

Supremum: structure of proofBearbeiten

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

  1. Prove that   is an upper bound of   : This is done showing   for all  .
  2. Prove that no number   is an upper bound of  : Take any   and show that there is a   with  .

Infimum: structure of proofBearbeiten

A proof that   is the infimum of a set   , could look as follows:

  1. Prove that   is a lower bound of   : This is done showing   for all  .
  2. Prove that no number   is a lower bound of   : Take any   and show that there is a   with  .

Maximum: structure of proofBearbeiten

the proof makes use of the definition of the maximum:

  1. Prove that   is an upper bound of  : This is done showing   for all  .
  2. Prove that  .

Minimum: structure of proofBearbeiten

In order to show that   is the minimum of   , you may proceed as for the maximum:

  1. Prove that   is a lower bound of  : This is done showing   for all  .
  2. Prove that  .

Examples for determining supremum/infimumBearbeiten

Finite setsBearbeiten

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.

Example (supremum and infimum of a finite set)

Consider the set  . Its maximum is   and its minimum  . The element   is included within the set and no element is greater than  . So  is the maximum. Analogously, the minimum is  .

As   is a maximum, it is also a supremum of   . And analogously,   is an infimum.

Exercise (for understanding): Determine the supremum and the infimum of the following sets:

  1.  
  2.  
  3.  

Solution:

  1. The supremum of   is its biggest element   . The infimum is the smallest element  .
  2. There is  . So the supremum is   and the infimum is  .
  3. There is  . So supremum and infimum of   are both given by  .

IntervalsBearbeiten

 
For each interval in the real numbers, the left boundary is the infimum and the right boundary is the supremum.

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:

Theorem (supremum and infimum of intervals)

Let   be an interval.So there are   with  , such that   takes one of the following 4 forms:

  1.  
  2.  
  3.  
  4.  

In that case,   is the infimum and   the supremum of the interval.

How to get to the proof? (supremum and infimum of intervals)

The above intervals differ in whether the endpoints  ,   are included or not. In any case, we know that for each   the following holds:  . So we know that   is a lower bound and   is an upper bound. That means,  ,   . We still have to show that   is indeed the greatest lower bound and   the smallest upper bound.

Let us assume that there is a  , so that   is also a lower bound. We can show that this leads to a contradiction:

To show that   cannot be a lower bound, we need to find an  , so that  . To construct such a  , we take the mean value between   and  , which by definition is greater than  . It could happen that   is so large that the mean value becomes larger than  . In this case,   is even larger than  , so any point of the interval is below   and   is not a lower bound. In any other case, the mean calue between   and   is an element of the interval and it is smaller than  . So   is not a lower bound in this case either. This is the contradiction, we have been looking for: we just showed that in any case,   cannot be a lower bound. So   is the greatest lower bound, i.e. the infimum. Showing that   is a supremum works by analogous arguments.

Proof (supremum and infimum of intervals)

Let   be an interval. It does not matter here, whether the boundary points are included or not. For each interval   , so   is a lower bound. We show that there is no greater lower bound than   by contradiction:

Assume that   also was a lower bound of  . The there is no  , such that  . We distinguish two cases:

Fall 1:   (small interval)

Let  . We can estimate

 

So   and hence  . By   we get

 

This contradicts the assumption of   being a lower bound of  .

Fall 2:   (large interval)

Now, take  . By our assumptions,

 

So   and we have  .

But since   we have  . Again, this contradicts   being a lower bound of  .

To-Do:

Add a figure depicting both cases.

In both cases, we get a contradiction to   being a further lower bound of   . So   is indeed the infimum of  .

The proof that   is the supremum is done analogously. We assume that   was an upper bound of  , which is lower than   . If we consider the number   for   and   for  , we will get a violation of this bound.

Alternatively, one may use the properties of suprema and Infiuma and apply the following trick: We use  .   is an infimum of   by the above arguments, so   is a supremum of  .

Exercise (for understanding): Determine the supremum and the infimum of the following sets:

  1.  
  2.  
  3.  
  4.   (here you may use, what you have learned in school about the sine function  )

  1. The supremum is   and the infimum is  .
  2. Since   , we have that  . This is an interval with infimum 1 and supremum 4.
  3.   is the set of all   with distance to   being less than   . That is an open interval  . Its infimum is   and the supremum is  .
  4.   is the set of all real numbers being covered by the sine function. The sine oscillates between   and  . So we just have an interval  . Hence, the infimum of   is   and the supremum is  .

Intervals of integersBearbeiten

Exercise (for understanding): Now, let us consider integer boundaries   with  . And we only pick out the integer numbers of the intervals:

  1.  
  2.  
  3.  
  4.  
What are supremum and infimum now? In which case do they even exist?

It helps a lot to know that the sets are all finite:

  1.  
  2.  
  3.  
  4.  

(Be careful with open boundaries! They remove one entire integer element.) In the first 3 cases, there is   and the set is non-empty.

  1.   and  
  2.   and  
  3.   and  

So there is always a supremum or an infimum. However, in the last case   can be empty. This happens exactly if  . In that case, there is neither a supremum nor an infimum. We only have the improper supremum   and the improper infimum  . However, those are no real numberes, i.e. no proper suprema/ infima. For  , the set is non-empty and we have supremum   and infimum  .

SequencesBearbeiten

 
The set  .

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:

Exercise (Sets of sequence elements)

What are the supremum and the infimum of the set  . Are they also a maximum or a minimum? Prove all your assumptions!

How to get to the proof? (Sets of sequence elements)

We follow a step-by-step approach:

Proof step: Visualize the set  .

The first elements of   are:

  1.  
  2.  
  3.  
  4.  
  5. ?

So the set   is of the form  , with all further elements approaching  .

Proof step: Give a hypothesis what may be the infimum or the supremum.

We see that the set is bounded from above by  . At the same time   is an element of the set, so   must be the maximum of the set. Furthermore, the set isbounded from below by  . Since the elements of the set are getting closer and closer to  , there can be no lower bound greater than  . It follows that   is probably the infimum of the set. Note that we are only making assumptions here because we are arguing intuitively. We still need a solid proof.


Proof step: Find a proof for the supremum / maximum.

We have already established that   is probably the maximum amount. So we have to show two things:

  •  
  •   for all  

The number   is definitely an elements of   , namely for   there is  . In order to prove that  is an upper bound of   , we need to show  . This can be achieved by equivalent transformations:

 

Now   is an obvious statement for natural numbers. But in the proof we must go the opposite way: Since we want to show the inequality  , we have to start at   and gradually transform this inequality into  . This is feasible because we have only used equivalent transformations above.

In the last chapter we have seen that every maximum of a set is automatically the supremum of the set (only the other way round it is not always the case). From this it follows that   is the supremum of  .


Proof step: Find a proof of the infimum / minimum.

To prove that   is an infimum, we have to show

  •   for all  
  • For all   there is an   with  

Um auch zu zeigen, dass   kein Minimum ist, haben wir außerdem zu beweisen, dass  . Zunächst muss ein Beweis for   for all   gefunden werden:

In order to show that in addition,   is not a minimum, we also have to prove that  . First of all, we have to find a proof to   for all  :

 

Now   is an obviously true statement, since   is positive. So in the later proof we can get to the inequality   starting from   by performing the above equivalent transformation backwards (i.e. adding   on both sides).

Let now continue with an arbitrary (possibly better bound)  . To rule out that this is a better bound, we have to find an   with   and   such that  . We choose here the variable   and not  , because we want to find a concrete element of the set   (in mathematics often   is used, if we are looking for a concrete  ). Let us reformulate this inequality to   to find a matching  :

 

Because of  , we have   and  . The Archimedian axiom now guarantees us that we can find a suitable  , because according to the Archimedian axiom the fraction   becomes smaller than any positive real number.

Finally, we need to show that  . Or put in different words,   is valid for all  . But because   there is   and therefore  . So this is rather simple. Now, we are ready to write down the proof:

Proof (Sets of sequence elements)

We prove that   is a maximum (and therefore supremum) of the set   and   is an infimum but not a minimum of the set  .

Proof step:   is a maximum of  

Proof step:   is an element of  

For   there is  . So  .

Proof step:   is an upper bound of  

For all   there is

 

So   is greater or equal to any element of  .

Proof step:   is the infimum of  

Proof step:   is a lower bound of  

For all   there is

 

So   is smaller or equal than any element of  .

Proof step: No other number greater than   can be a lower bound of  

Let   be arbitrary. Then,   and by the Archimedean axiom, there is an   with  . There is

 

Since   there is an element of   which is smaller than   . So   is no lower bound of  .

Proof step:   is not a minimum of  

There is   and hence  . So   is not an element of   and hence not a minimum.

Sets of function valuesBearbeiten

Exercise

Determine the supremum and the infimum of the set

 

How to get to the proof?

Let in the following be  . We proceed as follows:

Step 1: Visualize the set  .

The function   drawn in a diagram looks as follows:

The set   now consists of all values being hit by the function   , which is just the image of  .

Step 2: Make a hypothesis which numbers are the supremum and the infimum of the set.

We can assume that   is the supremum of  . Because  ,   is also hit by the function  . So this number is in   and should therefore be the maximum of this set.

Furthermore, the assumption that   is the infimum of   is obvious. The function always seems to be positive, i.e. greater or equal zero. The greater the absolute value of  , the closer the function values are to zero (at least that's how it looks at first glance). So   should be the infimum of  , whereas it is not directly in   and therefore it should not be a minimum.

Step 3: Find a proof for the supremum/ maximum.

We assert that   is the maximum of the set  . Because   we can directly see that  . Now all that is missing is the proof that   is an upper bound of the set  . For this we must prove that for all real numbers   we have the following inequality:

 

Let us transform this inequality by means of equivalent transformations:

 

We already know that the last inequality is fulfilled for all   . Since we have only used equivalent transformations, we can later recover the inequality  .

Step 4: Find a proof of the infimum / minimum.

Here we must first show that all elements of   are greater than or equal to zero. However,   is the quotient of two positive numbers, which must be positive again. All elements from   are therefore positive and hence greater than or equal to  .

Now, we need to show that   is also the largest lower bound of  . For this, let   be arbitrary. We have to find an element from   which is smaller than  . So there must be a   with

 

Let us transform this inequality by means of equivalent transformations:

 

In order to take the square root,   is required, i.e.  . But for the further proof it is no problem to assume  , because for   the last inequality is always fulfilled. The square number   is then always greater than the negative number  .

So consider  . Then:

 

For   we need to choose an   with  . This   automatically fulfils

 

hence   cannot by a lower bound of  .

Proof

We prove that   is a maximum and hence a supremum of   . Further, we prove that   is an infimum but not a minimum of  .

Proof step:   is the maximum of  

Proof step:   is an element of  

For   there is  . So  .

Proof step:   is an upper bound of  

There is

 

So   is an upper bound of  .

Proof step:   is an infimum of  

Proof step:   is a lower bound of  

There is for all  :

 

So 0 is a lower bound of  .

Proof step: No number greater than   is a lower bound of  

Let   be arbitrary. For   there is

 

for any real  , since   is negative then. For   choose   such that   . Then the following inequality is satisfied.

For any   there is at least one real number   with  . For real numbers of this kind we have

 

So   cannot be a lower bound of  , which proves that   is the largest lower bound of  .