Question about supremum and infimum

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In summary, if a is the supremum of a set {a1, a2, a3, ...}, then there must be some element a_n in the set such that the absolute value of the difference between a and a_n is less than any given epsilon. This is because a, being the least upper bound, must have an element in the set that is arbitrarily close to it. This is also true for finite sets.
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hkcool
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Say we have [itex]a = \sup \{ a_{1}, a_{2}, a_{3}, ... \}[/itex]. Then does this mean we can find some [itex]a_{n} \in \{ a_{1}, a_{2}, ... \}[/itex] such that

[tex]|a - a_{n}| < \varepsilon[/tex]

? My reasoning is that since a (the supremum) is the least upper bound of the set, we have to be able to find some member of the set that is arbitrarily close to a otherwise a wouldn't be a supremum anymore. Is this true?
 
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  • #2
For any ε > 0 then yes, that's true.
 
  • #3
Well, if the the sup in the set iself then epsilon can indeed be zero.
You may also find it interesting to think about finite sets.
 

1. What is the definition of supremum and infimum?

The supremum of a set is the smallest upper bound, while the infimum is the largest lower bound. In other words, the supremum is the lowest possible value that is still greater than all the values in the set, and the infimum is the highest possible value that is still less than all the values in the set.

2. How do you find the supremum and infimum of a set?

To find the supremum and infimum, you need to first determine the upper and lower bounds of the set. Then, the supremum is the smallest of the upper bounds, and the infimum is the largest of the lower bounds.

3. Can a set have both a supremum and infimum?

Yes, a set can have both a supremum and infimum. This occurs when the set has a finite number of elements, and the highest value is equal to the supremum and the lowest value is equal to the infimum.

4. What is the difference between supremum and maximum, and infimum and minimum?

The supremum and infimum are bounds on a set, while the maximum and minimum are actual values in the set. The supremum and infimum may or may not be included in the set, while the maximum and minimum must be included in the set.

5. How are supremum and infimum used in analysis and mathematics?

Supremum and infimum are important concepts in analysis and mathematics as they help to define important properties of sets, such as compactness and continuity. They are also useful in proving the existence of solutions to optimization problems and in defining the convergence of sequences and series.

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