[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

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

This is both the reason why mathematics wants formal proofs instead of heuristics, intuition or induction, as well as why it sometimes is incredibly hard to really proof things.

Take f(x) = x, as it is the simplest example. “y goes to infinity” means that y gets at some point and onward larger than any finite bound. Then “f(x) goes to infinity when x goes there” is almost tautological, as it says: “whenever x gets and stays larger then any finite bound, then x gets and stays larger then any finite bound”. Or without the fluff “if A then A”, which is absolutely always true.

Now for other functions like x² , e^x , log(x) , or x+100·sin(x) the argument might be more convoluted. Lets try the last one:

“Whenever x gets and stays larger then any finite bound, then x+100·sin(x) gets and stays larger then any finite bound”.

We can note that sin(x) is between -1 and 1, therefor we know that x+100·sin(x) is at least as big as x-100. Looking as x-100 as a kind of “worst case” simplifies things:

“Whenever x gets and stays larger then any finite bound, then x-100 gets and stays larger then any finite bound”.

No more sine to deal with! If we now want x-100 to be (and stay) larger than a finite bound (lets say 1,000,000), then we guarantee that by choosing x itself to be bigger than that bound plus 100 (here: 1,000,100). This proves what we wanted!

Never did we have to check an infinite number of cases by hand, nor is there any wiggly room, despite x+100·sin(x) wiggling up and down all the time (draw a graph if you want to see this). We argued purely by reason and logic, which goes further than simple calculation.

For some way questions, showing that something grows without bound can become seriously complicated; too complicated for even modern mathematicians. Only time will show if we are able to absolutely prove such things, but the chance for failure is always there. Heck, it is even an unsolved problem if there is a way to always and definitely decide (an “algorithm”) if two elementary expressions describe the same function.