I have recently finished writing up my notes on bounded sets in Real Analysis (I will be posting my incomplete notes for Real Analysis shortly), and what is fascinating me was how the concept of these sets led to the formation of limits. Here are my thoughts:
Suppose that is an ordered field and is bounded above in . A number is called a least upper bound of if (i) is an upper bound for , and (ii) if is any upper bound for , then Similarly, if is bounded below in , then a number is called a greatest lower bound of if (i) is a lower bound for , and (ii) if is any lower bound for , then From this definition, we can see that the set is a bounded set both above and below, and is a subset of ordered field . We can think of lub and glb as the closest approximation of a value that cannot be defined. The reason for having the least and greatest bounds of a set is to not find the value of ; however it is to find a way to get close enough to . And that is essentially what a limit is.
If we have a limit from to () for some function , we are relatively finding the glb closest to . We can then have an intuition for the limit . This shows that we want to get as close to as possible. From here, these infinite limits are classified as sequences. However, I will leave the details for you to study on your own.