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 th element of the modified diagonal, by definition, is, e.g., , and the th element of the th line is . So, the sequence is different from the th line at least in the th element (maybe in other places as well), and this is true for all .