What are upper bounds and supremums in proofs?

[Miike012]

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?

 

[Joffan]

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 $$\frac{1}{1+e^{-x}}$$ 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