The Proof of Uncountable Infinity

The proof is that there's no way to assign all of those numbers between zero and one to the rooms in such a way that no one misses out on a room. If you list all the people in what rooms they're going into, you can then generate a new number. But once you've got a decimal point, you can have infinite combination of numbers. With whole numbers, it's just. Yes, but whole numbers can also get as large as you want. There's still as many digits as you fancy in one direction at the decimal point as there are on the other direction.