Essential supremum and essential infimum

Alright kiddo, let me explain essential supremum and essential infimum in a way that's easy for you to understand.

Imagine you have a bunch of toys (or cookies, if you like). Some toys are bigger than others, and some cookies are sweeter than others. Now, of all the toys you have, what's the biggest toy you have? That toy is called the "supremum" of all your toys.

Similarly, of all the cookies you have, what's the sweetest cookie you have? That cookie is called the "supremum" of all your cookies.

But sometimes, you might have some "bad" toys – toys that are broken, for example – or some "bad" cookies – cookies that are stale, perhaps. So, we might ignore those toys or cookies, and only consider the good ones.

This is where the concept of "essential supremum" and "essential infimum" comes in.

Let's say you have a bunch of toys, including some bad ones. Of all the good toys you have, what's the biggest one? That toy is called the "essential supremum" of your toys. It's like asking what's the biggest toy you can actually play with, without considering the broken ones.

Similarly, if you have a bunch of cookies, including some bad ones, of all the good cookies you have, what's the sweetest one? That cookie is called the "essential supremum" of your cookies. It's like asking what's the sweetest cookie you can actually eat, without considering the stale ones.

You can also do the same thing with the smallest toys or cookies, which is called the "essential infimum". It's like asking what's the smallest toy you can actually play with, or what's the least sweet cookie you can actually eat, without considering the bad ones.

So, the essential supremum is like finding the best of the good things, and the essential infimum is like finding the worst of the good things. And that's what essential supremum and essential infimum mean!