Archimedean ordered vector space

Imagine you have a bunch of arrows, where each arrow has a length and a direction. These arrows represent numbers, and they live in something called a vector space. Now, let's say we have a special rule that says these arrows must always have a certain order. This means that if we have two arrows A and B, we know which one is "bigger" or "smaller" based on their lengths and directions.

This is what we call an ordered vector space. But there's more! Let's say we have a third arrow, C. This arrow is somewhere between A and B in terms of size. We could say that C is the average of A and B.

So, in summary, an archimedean ordered vector space is a special kind of vector space where the arrows (numbers) have a specific order, and we can find the average (or what we call a limit) of any two arrows that are close together. It's like playing with number arrows, but with extra rules to follow!