Quote Originally Posted by stribor40 View Post
When you have numbers...

0, 1, 2, 3, 4, 5, 6, 7, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
2, 3, 4, 5, 6, 7, 8, 9, ...
3, 4, 5, 6, 7, 8, 9, 10 ...

How does chosing something different from diagonal (0,2,4,6..) to something 1,5,7,9...ensuring that is different from everything else in the list (regardless of being repeating or non repeating)
Well, is 1, 5, 7, 9, ... different from the first line? Is it different from the second, third, fourth lines? The answer to why the modified diagonal is different is because we modify it to be different. In the notation of post #13, the nth element d_n of the modified diagonal, by definition, is, e.g., x_{n,n}+1, and the nth element of the nth line is x_{n,n}. So, the sequence d is different from the nth line at least in the nth element (maybe in other places as well), and this is true for all n.