Cantor's Diagonal
You can list the natural numbers: 1, 2, 3, 4, and so on forever. You can list the integers, if you alternate: 0, 1, −1, 2, −2, 3, −3, … You can list the rationals, even, by walking through them in a clever zigzag. It feels like you should be able to list anything, given infinite time and a good enough scheme.
But the real numbers between 0 and 1 cannot be listed. No scheme works. No matter how you try to enumerate them, you will always miss some — in fact, you will miss almost all of them.
The proof, due to Cantor in 1891, is short. It takes the form of a recipe: hand me any list, and I will hand you back a number that isn't on it.
The recipe
Suppose someone gives you an infinite list of real numbers between 0 and 1. They claim every such number appears somewhere on the list. You want to construct one that doesn't.
Look at the first number's first digit after the decimal point. Write down a digit different from it.
Look at the second number's second digit. Write down a digit different from it.
The third number's third digit. The fourth's fourth. The Nth's Nth. Down the diagonal.
The number you've written differs from the Nth number in (at least) its Nth digit, so it can't equal the Nth number. For any N. So it isn't on the list.
"Just add it to the list"
The natural objection: fine. Take the missing number and add it to the list. Now it's on the list, problem solved.
But the recipe still works. Run it again on the new list. You'll get a different number, also missing.
You can do this forever. The list never closes. Every list of real numbers, no matter how cleverly constructed, has uncountably many holes.
What this shows
There is no way to put the real numbers in one-to-one correspondence with the natural numbers. The reals are uncountable: a strictly larger infinity than the integers or the rationals.
This was a shock when Cantor published it. It was the first proof that infinity comes in sizes — that there are strictly more real numbers than natural numbers, and no clever counting will ever close the gap.
The argument is one page. The implication keeps unfolding for over a century.