Cantor's diagonal argument

one of my favorite proofs, also from my favorite chapter of Gödel, Escher, Bach.

say you supposedly have a set of all irrational numbers, some thing like the following;

$\mathbb{N}$	$\mathbb{R}$
1	0.5823748
2	0.8583252
3	0.3749182
4	0.9284756
5	0.1472938
6	0.6382194
7	0.8583252
…	…
n	0.3862767

we can construct n, a number that doesnt exist in the set of all irrationals by:

1. start first column of first row, and choose a number that is NOT that number
2. second column second for, choose a number that is NOT that number
3. repeat until the end of the list
4. you have a new item $n$ that is NOT in the list

if you add it to the list, you can now construct another $n$ that still isnt in the list .