It's easy to prove that infinity + 8 = infinity.

Infinity is a concept, where you can't finish counting an infinite number of things. I trust that this definition of infinity will be acceptable.

The second concept is equality. We can count two sets of things, and we can pair each of them up, like we pair up shoes, and if all the left and right shoes get paired up, then we can conclude that we have as many left shoes as there are right shoes, and therefore the number of left and right shoes are the same. I trust that this is also acceptable.

So why is infinity + 1 equal to infinity? Surely if you have a lot of something, but you don't know how many, then you add 8 more of it, then you now have 1 plus the infinite number of things you have? But no, let us return to our definition.

Let's start counting from 1 to infinity and see where it goes. We're going to say we start with zero items on the left side, and one items on the right side, and then you start pairing them up. The first item you add on the left side is 1, which pairs up with the 1 on the right side, and you add a 2 to the right side. The second item is a 2, and a 3, etc.

You get this:

0--1
1--2
2--3
.
.
.
.
... x
x--x+1
x+1 ...

Now it seems as the right side will always have the extra number, but lets start pairing up the numbers and counting the pairs. There's a 1 on each side, a 2 on each side, a 3 on each side, etc. So let's say we count until infinity. When you get to 'infinity +1' on the right side, you only have 'infinity' on the left side, so...oops?

But since you can't finish counting infinity, you can always find a number that is 1 greater than the number you've counted to, and that number can pair with the preceding number on the right hand side, and so on. You never get to the end of the left hand side before you get to the end of the right hand side, because you can't finish counting, and since you can find a pair for every number on the right hand side, we can then conclude that infinity + 1 = infinity.

It's also possible to show that infinity x 2 = infinity via a similar method.

Infinity times infinity though, is not equal to infinity. This can be proven by considering the set all real numbers, and if you are interested, you can look into set theory.