Proof:
Try to list out all numbers (rational or not) and have them in a column. For rational numbers, include the trailing zeros.
E.g:
.378213460000000000...
.823479879283458237...
.283789125903045721...
.129039784823487952...
And so on. You have listed a countable number of real numbers between zero and one. (They're in a list!)
However! I can give you a number between zero and one that you did not list! For the nth digit of my number, I will add one to the nth digit of your nth number. (Nines go to zeros)
Using the example above, I would be choosing .4341...
Since this number differs from each number in your list, it is a number between zero and one that you don't have in a one-to-one correspondence with the natural numbers. Thus, the quantity (cardinality) of real numbers between zero and one is greater than the quantity (cardinality) of natural numbers.
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First Delver! (I was the first non-tester/dev
to conquer TCB.)
d
/dy