Is the Equation sup(X) = -inf(-X) Correctly Proven?

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This is because for 1/b to be bounded, it must not approach infinity or negative infinity. Therefore, b must be greater than or equal to 1 in absolute value. In summary, in order for 1/b to be bounded, b must be non-zero and lie within the range (-∞,-1] U [1, ∞) with a minimum absolute value of 1.
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peripatein
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Hello,

(1) In order to prove that sup(X)=-inf(-X), for XC=R and -X={x E R|-x E X}, does it suffice to write:

sup(X): for every x E X, x <= M, M E R
inf(-X): for every x E -X, -x >= m ⇔ x <= -m, m E R

Hence, m=min(-X) and M=max(X) as they are both part of R and as there could be only one supremum M must be equal to -m.

Is this a valid proof?

(2) Which conditions need b to fulfill so that 1/b is bounded, for b E R? I have managed to arrive at |b|>=1. Are there any other requirements, conditions?
 
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Yes, in order for 1/b to be bounded, b must be non-zero and b must also lie within the range (-∞,-1] U [1, ∞).
 

What is the proof for Sup(X)=-inf(-X)?

The proof is based on the definition of supremum and infimum. Supremum (Sup) is the least upper bound of a set of numbers, while infimum (Inf) is the greatest lower bound. In other words, Sup is the smallest number that is greater than or equal to all the numbers in the set, while Inf is the largest number that is less than or equal to all the numbers in the set.For Sup(X)=-inf(-X) to hold, it means that the smallest number that is greater than or equal to all the numbers in the set X is equal to the largest number that is less than or equal to all the numbers in the set -X.

Why is it important to prove Sup(X)=-inf(-X)?

It is important to prove Sup(X)=-inf(-X) because it is a fundamental property of real numbers that helps us understand the behavior of sets. It is also useful in various mathematical proofs and can be used to solve optimization problems. Moreover, understanding this proof can help us gain a deeper understanding of the concept of supremum and infimum.

What is the intuition behind Sup(X)=-inf(-X)?

The intuition behind Sup(X)=-inf(-X) is that the smallest number that is greater than or equal to all the numbers in the set X must be equal to the largest number that is less than or equal to all the numbers in the set -X. In other words, the upper bound of a set is equal to the lower bound of its negative counterpart.

Can you provide an example to illustrate Sup(X)=-inf(-X)?

Yes, for example, let X = {1, 2, 3}. The Sup(X) = 3, since 3 is the smallest number that is greater than or equal to all the numbers in X. Now, the negative counterpart of X, -X = {-1, -2, -3}. The Inf(-X) = -3, since -3 is the largest number that is less than or equal to all the numbers in -X. Therefore, Sup(X) = -Inf(-X).

Are there any exceptions to Sup(X)=-inf(-X)?

Yes, there are exceptions to Sup(X)=-inf(-X). This proof only holds for non-empty sets with finite supremum and infimum. If the set X is empty or has an infinite supremum or infimum, then this property does not hold. Also, if the supremum and infimum of X are equal, then Sup(X)=-inf(-X) is not valid.

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