Prove that inf S = - sup {-s: s in S}

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In summary, "inf" and "sup" refer to the infimum and supremum of a set, respectively. The infimum is the greatest lower bound of a set, while the supremum is the least upper bound. To prove that inf S = - sup {-s: s in S}, we show that the infimum of S is equal to the negative of the supremum of the set {-s: s in S}. An example of this equation is when S = {1, 2, 3}, where inf S = - sup {-s: s in S}. This equation has practical applications in mathematical analysis, optimization, economics, and finance.
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Homework Statement



Let S be a nonempty subset of R that is bounded below. Prove that inf S = - sup {-s: s in S}

The Attempt at a Solution



Let a0 = inf S. Thus, for all s in S, a0 is less or equal to s; or -a0 greater or equal to -s.
If u is any upper bound for -S, u is greater or equal to -a0; or -u less or equal to a0.

and here I have difficulty proceeding further. I don't quite see the logic anymore :-(

Any help is greatly appreciated.


 
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Your attempt at the solution is on the right track. Let me guide you through the rest of the proof.

First, recall that the infimum of a set S is the greatest lower bound of S. This means that a0 is the largest number that is less than or equal to all the elements in S. In other words, for any s in S, a0 is less than or equal to s.

Now, let's consider the set {-s: s in S}. This set is simply the set S with all its elements negated. So, if s is in S, then -s is in {-s: s in S}. This means that -s is also an element of S.

Since a0 is the greatest lower bound of S, it must also be less than or equal to -s for all s in S. In other words, -a0 is the greatest upper bound of the set {-s: s in S}. This means that -a0 is the supremum of {-s: s in S}.

Therefore, we can conclude that a0 = inf S = - sup {-s: s in S}. I hope this helps clarify the logic for you. Good luck with your studies!


 

1. What does "inf" and "sup" mean in this context?

"inf" stands for infimum, which is the greatest lower bound of a set. "sup" stands for supremum, which is the least upper bound of a set.

2. What is S in this equation?

S is a set of real numbers.

3. How do you prove that inf S = - sup {-s: s in S}?

To prove this equation, we need to show that the infimum of S is equal to the negative of the supremum of the set {-s: s in S}. This can be done by first showing that the infimum of S is a lower bound for the set {-s: s in S}, and then showing that the negative of the infimum is an upper bound for the set {-s: s in S}. By definition, the infimum is the greatest lower bound, and the negative of the supremum is the greatest upper bound. Therefore, if the two are equal, they must be equal to each other.

4. Can you provide an example to illustrate this equation?

Let S = {1, 2, 3}. The infimum of S is 1, and the supremum of {-s: s in S} is -3. So, inf S = 1 and - sup {-s: s in S} = -(-3) = 3. Therefore, inf S = - sup {-s: s in S}.

5. What are some real-life applications of this equation?

This equation is commonly used in mathematical analysis and optimization problems. It can also be applied in economics and finance, where it is used to find the optimal value in a given set of values.

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