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This "orthodox" maths is not a good approach to the subject since it actually stops you counting correctly. The ultimate foolishnes is taking a finite set of counting numbers, O(N), and doing a one-to-one mapping (bijection) to the corresponding set of rational numbers, O(N²). As you consistently increase N to infinity it is obvious that the mapping gets worse and worse, and yet somehow we are asked to believe that infinity somehow makes it all ok. This is not the correct way of dealing with divergent series.

Here is a little problem designed to restart the thinking process:

And then it is time to re-learn the subject:

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