[ELI5] how do we prove that something goes to infinity or – infinity since we can’t observe it?

Question

For instance how do we know that something weird doesn’t happen at some insanely large number?

In: 3

Answer 1

You say a sequence goes to infinity if for any number you pick, the sequence has a larger number.

So let’s use natural numbers, 1,2,3,4….

You pick a any number you want, say 999. I can then say at position 1000, in the sequence is 1000 which is greater than 999.

You would generalise it to be for any number e, there is a number e+1 in the sequence that is greater than it. So no matter what number you can think of, there is always a number in the sequence greater than it

Answer 2

You say a sequence goes to infinity if for any number you pick, the sequence has a larger number.

So let’s use natural numbers, 1,2,3,4….

You pick a any number you want, say 999. I can then say at position 1000, in the sequence is 1000 which is greater than 999.

You would generalise it to be for any number e, there is a number e+1 in the sequence that is greater than it. So no matter what number you can think of, there is always a number in the sequence greater than it