Piling-up lemma

Okay kiddo, are you ready? Today we are going to talk about something called the “piling-up lemma”. It’s like stacking up your building blocks, but with math!

You know how you can add up numbers to get a bigger number? Well, in math, we can do something similar with functions. You just take one function and you add another function to it. The new function is the sum of the two functions.

The piling-up lemma is like a rule that says if you have a bunch of functions added together like this, and they're all positive, then their sum is going to be positive too. It's like adding up a bunch of things you like - such as toys, candies, or pizza slices - and you end up with something even better!

This might sound simple, but it can be really useful in math. For example, if you’re trying to figure out the area under the curve of a function, you can split it up into smaller parts and add them all up using the piling-up lemma.

But wait, there’s more. The piling-up lemma also works for negative functions too! If you have a bunch of negative functions added together, then the sum of them is going to be negative too.

Let’s use an example to help you understand better. Let's say you have a cake and you cut it into many pieces. Each piece is a function representing how much cake is there in that part. The piling-up lemma says that no matter how you stack up the pieces, the total amount of cake you get is going to be positive, because it's still the same cake!

So that's the piling-up lemma, kiddo. It's a rule that helps us add up functions, and it's like stacking up toys or building blocks. We can use it to find the area under the curve and it works for both positive and negative functions. Now go play with your building blocks, and remember to pile them up!