The Supremum and Infimum
Imagine, in an abstract way, that this set A that we just talked about, is.. your house! In it are all elements, things that are part of your house.
Now, let's bring back the concepts we talked about before, the upper and lower bounds.
In this case, your upper bound would be the roof of your house and everything else above it. The sky, the atmosphere and up to the ends of the universe.
Down below the floor, is the foundation. The ground the house lies on.
You can go anywhere below, the ground, the core of the earth, disappear into another universe, that's all lower bounds of this house.
Now imagine that we had a magic wand that could somehow make your house taller either upwards or downwards. In mathematics, this would be a function.
You would be able to raise and raise the roof to anything you want that this transformation can do. If it can't make it any taller, the roof at that point is the highest possible it can be.
The same goes for downwards. You keep expanding and expanding underground, if there is no limit, great!
If your magic wand can only go so far, then the floor at that point is also the lowest possible it can be.
Where is this going?
Now, right at that point, you have the start of the next. The roof touches the air, the lowest upper bound. The floor, the ground. The highest/greatest lower bound.
These are the concepts of The Supremum and Infimum.
The Supremum of a set A, denoted by
Similarly, The Infimum of a set A, denoted by
The concept of a set or function being "squeezed" between these boundaries or only existing within those is a concept that will be extremely useful soon.
In order to lay down some bases for new concepts, the next topic will be:
Introduction to Sequences