What a proof is, and why checking a million cases is not one
A million examples is not proof. One counterexample is disproof.
In almost every subject, knowing means having very good evidence. In mathematics it means something stronger: a statement is proved when it follows by logic from things already accepted, and once proved it does not get overturned. Pythagoras' theorem was true 2,500 years ago and no future discovery will affect it.
That is unusual enough to be worth understanding properly.
Why examples are not enough
Consider: is n^2 + n + 41 always prime? Try it. n=0 gives 41, prime. n=1 gives 43, prime. It keeps working - for every value from 0 to 39.
At n=40 you get 40^2 + 40 + 41 = 1681 = 41 x 41. Not prime.
Forty consecutive confirmations, then failure. This is why mathematicians are unmoved by 'it works for everything I tried'. It is also why proof by contradiction and proof by induction exist - to cover infinitely many cases at once, which no amount of checking can do.
The main kinds
Direct proof
Start from what you are given, apply known rules, arrive at the conclusion. To show the sum of two even numbers is even: an even number is 2k for some whole k, so 2a + 2b = 2(a+b), which is 2 times a whole number, so even. Done - and done for all even numbers, not a sample.
Proof by contradiction
Assume the opposite of what you want. Follow it honestly. Arrive at something impossible. Since the logic was valid, the assumption must have been false.
Euclid's proof that primes never run out works this way, and so does the proof that the square root of 2 cannot be written as a fraction - which reportedly disturbed the Pythagoreans enough that they tried to suppress it.
Proof by induction
For statements about every whole number. Show it holds for 1. Then show that if it holds for n, it must hold for n+1. Those two facts together cover every number, because 1 gives 2, which gives 3, and so on.
The usual image is dominoes: knock the first, and guarantee each knocks the next, and you have accounted for all of them without pushing each one.
Two short checks establish a statement about infinitely many numbers. That leverage - finite work, infinite conclusion - is the thing induction is for.
Counterexamples do the opposite job
To prove a general claim you must handle every case. To disprove one you need exactly one failure.
That asymmetry is worth carrying around. 'All swans are white' needed centuries of white swans and survived; one black swan in Australia ended it permanently. Finding a single counterexample is often far more efficient than attempting a proof, and checking for one first is good practice.
This is also how debugging works. 'The function works' needs every input to behave. 'The function is broken' needs one input that misbehaves. Which is why a single reproducible failing case is worth more than any amount of testing that passed.
Why it is worth caring about
Most people will never write a formal proof after school. The habit underneath it transfers anyway: distinguishing 'I have seen this work several times' from 'this must work', noticing when an argument covers all cases rather than the convenient ones, and knowing that one solid counterexample beats a pile of supporting examples.
That is a general-purpose defence against being wrong, and it came out of a subject that seems, from a distance, to be about numbers.