Georg Cantor has showed there are more real numbers than natural numbers in his diagonal argument. Assuming that two sets have the same size if we can make a pair up elements from set A with elements from set B. Now if we make a list of natural numbers, then no matter what list of real numbers we provide, there will always be another number that's not on the list (created by adding '1' to diagonal of the list, for example).
Now, it's a proof that there are more reals. But what if I told you, before Cantor, that there are equally many reals and natural numbers, because we can make a list of natural numbers and assign a real number to each natural number? I will tell you, look, it can't be the case there are more real numbers, because there are infinitely many nautral numbers and you will never run out of natural numbers to pair them up with real numbers!
Would it be considered a valid proof? Actually, wasn't it considered a valid proof until Cantor came up with his theory? What if there are other such theorems in mathematics commonly believed to be true, with accepted proofs, and some day someone will come up with a counterexample, disproving them? Does it mean we can never be sure if a mathematical proof is valid?