List every real number between 0 and 1 — one is guaranteed to be missing
Suppose you claim to have listed every real number between 0 and 1, pairing the first with 1, the second with 2, and so on forever, so no such number is left off your list. Georg Cantor showed how to build a brand-new number by looking down the list and, digit by digit, always choosing a different digit than the one sitting on the diagonal. Could this new number already be sitting somewhere on your supposedly complete list?
Reveal the answer
No — the constructed number differs from the nth number on the list in at least its nth digit, for every n, so it can't match any entry, no matter how the list was built. Georg Cantor published this diagonal method in 1891, giving a second, sharper proof that the real numbers are uncountably infinite — strictly "more numerous" than the whole numbers.