First off, we have to know how to COUNT the integers, or at least how to 'count' a set that may be as large as the integers.
Makes no sense! you may cry. Except Cantor showed us how to.
He said that two sets contain the SAME number of elements if they can be matched one-to-one with none left over.
Hence integers 1,2,3,4,5,6.....n,......... can be matched with the even numbers 2,4,6,8,10,......2n,....... in an obvious way.
They can also be matched with the rational numbers, since each rational number can be written as p/q (no common factor, both integers). Imagine the rationals laid out one a square grid, with p/q living at p on the x-axis, q on the y-axis. Then imagine traversing ALL the sites on this grid by following a spiral path starting at the origin. You WILL visit every such number, eventually, and the matching is just the order in which you visit them.
Hence as many integers as even numbers, and as many rationals as integers.
Now, lets try and be extra clever, and try and count all numbers between 0 and 1, which mat have finite or infinite decimal expansions. We'll try and do the matching, and make a list, which we will state contains ALL such decimals between 0 and 1. So we might have:
1: 0.345
2: 0.0679876555
3: 0.333333333333333333333.........
4: 0.000000910000000000000........
and so on.
but now a create a new number. I take the first digit of the first number in my list, and choose a different value. I take the second digit of the second number in my list, and choose a different value, and so on. These digits form my new number.
So, 1st digit can't be 3 -- I'll choose 7, to give 0.7
2nd digit can't be 6, I'll choose 0, to give 0.70
3rd digit can't be 3, I'll choose 9, to give 0.709
and so on. I hence create a new decimal which is DIFFERENT to every number in my list, by construction. Hence such a list can never be complete.
This means that you can't do a one-to-one match between deciamls between 0 and 1, and the integers. There are always decimals not in the list. Hence the number of decimals is larger than the infinity of integers.
This argument is called Cantors diagonal slash, and was also used by Godel in his proof of the first Incompleteness Theorem.