No Stupid Questions

There are many structures of proof. A simple one might be to prove a statement is true for all cases, by simply examining each case and demonstrating it, but as you point out this won't be useful for proving statements about infinite cases.

Instead you could assume, for the sake of argument, that the statement is false, and show how this leads to a logical inconsistency, which is called proof by contradiction. For example, Georg Cantor used a proof by contradiction to demonstrate that the set of Natural Numbers (1,2,3,4...) are smaller than the set of Real Numbers (which includes the Naturals and all decimal numbers like pi and 69.6969696969...), and so there exist different "sizes" of infinity!

For a method explicitly concerned with proofs about infinite numbers of things, you can try Proof by Mathematical Induction. It's a bit tricky to describe...

• First demonstrate that a statement is true in some 1st base case.
• Then demonstrate that if it holds true for the Nth case, then it necessarily holds true for the (N+1)th case (by doing some clever rearranging of algebra terms or something)
• Therefore since it holds true for the 1th case, it must hold true for the (1+1)th case = the 2th case. And since it holds true for the 2th case it must hold true for the (2+1)=3th case. And so on ad infinitum.

Wikipedia says:

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.

Bear in mind, in formal terms a "proof" is simply a list of true statements, that begin with axioms (which are true by default) and rules of inference that show how each line is derived from the line above.

As you said, we have infinite numbers so the fact that something works till 4x10^18 doesn't prove that it will work for all numbers. It will take only one counterexample to disprove this conjecture, even if it is found at 10^100. Because then we wouldn't be able to say that "all" even numbers > 2 are a sum of 2 prime numbers.

So mathematicians strive for general proofs. You start with something like: Let n be any even number > 2. Now using the known axioms of mathematics, you need to prove that for every n, there always exists two prime numbers p,q such that n=p+q.

Would recommend watching the following short and simple video on the Pythagoras theorem, it'd make it perfectly clear how proofs work in mathematics. You know the theorem right? For any right angled triangle, the square of the hypotenuse is equal to the sum of squares of both the sides. Now we can verify this for billions of different right angled triangles but it wouldn't make it a theorem. It is a theorem because we have proved it mathematically for the general case using other known axioms of mathematics.

https://youtu.be/YompsDlEdtc