Understanding Upper Bound & Sup in Theorem Proving

In summary, "upper bound" is a term used in proving theorems and refers to any value that is greater than or equal to all the values in a set. "Sup" is short for supremum, which is the least upper bound of a set of numbers. It is not necessarily a member of the set, but it is an upper bound.
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Miike012
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This calc book that I am reading uses words like "upper bound" and "sup" a lot when proving theorems. I have never heared these terms before so it makes it hard for me to understand the proofs.

I think it has to deal with max's values of a graph: For example given a set S of all elements c in a ≤ c ≤ b would the the upper bound in the following graph in [a,b] be c? and also c = Sup S...
is this correct?
 

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Upper bound is any value that is greater than or equal to all the values in a set.

sup is short for supremum, the least upper bound of a set of numbers. The range of a function might be such a set. The supremum is not necessarily a member of the set: for example, the supremum of the range of the sigmoid function [tex]\frac{1}{1+e^{-x}}[/tex] is 1, but that is not a value of the function. 1 is an upper bound to this function, but so are 2, 100 and 100!2
 

1. What is an upper bound?

An upper bound is a mathematical concept that refers to the maximum value that a set of numbers or variables can have. It serves as an upper limit for the values in the set.

2. What is the significance of an upper bound in theorem proving?

In theorem proving, an upper bound is used to determine the maximum value that a variable can take on in order for a theorem to hold true. It helps to narrow down the possible values and make the proof more efficient.

3. How is an upper bound related to the concept of supremum?

The supremum, or "sup" for short, is the smallest upper bound for a set of numbers or variables. It is the highest possible value that the set can have.

4. Can you give an example of using an upper bound and sup in theorem proving?

For example, in proving that the sequence (n^2 + 1) is bounded above, we can use an upper bound of 10. This means that for all values of n, the sequence will never exceed 10. The supremum in this case would be 10, as it is the smallest upper bound.

5. Are there any limitations to using upper bounds and sup in theorem proving?

While upper bounds and sup can be useful tools in theorem proving, they are not always applicable. In certain cases, the values in a set may not have a maximum value, making it impossible to find an upper bound. Additionally, the use of sup can be limited by the complexity of the problem and the available computational resources.

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