Definition of supremum and infimum using epsilons ?

In summary, the conversation discusses the definition of an infimum and the importance of upper bounds in relation to it. The initial statement is not a complete definition as it does not account for the possibility of strict inequality. The conversation also references a theorem stating that if L is a lower bound for a set A in R, then L = inf A if there exists an x in A with x - L < epsilon for every epsilon > 0.
  • #1
AxiomOfChoice
533
1
Is this what it is:

"For every [itex]\epsilon > 0[/itex] there exists [itex]x\in A[/itex] such that [itex]x \leq \inf A + \epsilon[/itex]."

...and similarly for the supremum?
 
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  • #2
No, that's not quite it. Any number y which is greater than all numbers in A would satisfy your definition.
 
  • #3
Well in the first definition the OP said "there exists", so really the concern of upper bounds isn't really important. This is a useful proposition, and sometimes the inequality is strict, though that doesn't matter all that much.
 
  • #4
What "first definition"? Are you talking about something other than:

"For every [tex]\epsilon > 0[/tex] there exists [tex]x\in A[/tex] such that [tex] x \leq \inf A + \epsilon .[/tex]"?

While that is true about the infinimum, it won't do for the definition for the reason I gave.
 
  • #5
LCKurtz said:
What "first definition"? Are you talking about something other than:

"For every [tex]\epsilon > 0[/tex] there exists [tex]x\in A[/tex] such that [tex] x \leq \inf A + \epsilon .[/tex]"?

While that is true about the infinimum, it won't do for the definition for the reason I gave.

That's correct. We must also state that inf A is a lower bound.
 
  • #6
Yeah I realized I was thinking of the theorem that states that if L is a lower bound for a set A in R, then L = inf A iff for every epsilon > 0, there is an x in A with x - L < epsilon. My apologies, of course it's not a definition.
 

1. What is the definition of supremum?

The supremum of a set of numbers is the smallest number that is greater than or equal to all the numbers in the set. It is also known as the least upper bound.

2. What is the definition of infimum?

The infimum of a set of numbers is the largest number that is less than or equal to all the numbers in the set. It is also known as the greatest lower bound.

3. How is epsilon used in defining supremum and infimum?

Epsilon (ε) is used as a parameter in the definition of supremum and infimum in order to provide a precise value that is close enough to the actual supremum or infimum. The difference between the supremum/infimum and the epsilon value is known as the approximation error.

4. What is the role of epsilon in proving the existence of supremum and infimum?

Epsilon serves as a tool for proving the existence of supremum and infimum. By using epsilon, we can show that there is a number in the set that is close enough to the supremum/infimum, and thus, prove its existence.

5. How is the definition of supremum and infimum using epsilon useful in real-world applications?

The definition of supremum and infimum using epsilon is useful in many real-world applications, such as optimization problems, economics, and engineering. It allows us to find the best possible solution or the smallest/largest value that satisfies certain conditions. It also helps us to approximate values and make decisions based on precise calculations.

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