If $A_1, A_2, ...$ is a list of countable sets, then$A_1 \cup A_2 \cup A_3 ...$ is also countable?

Question

Does anyone know the reasoning behind the diagonal ordering in the second picture? I have seen it before, but am not sure what it's showing relating to this proof.

Answer 1

Do you need any reason further than just "it does the job!"? (It collects every element of every list exactly once, in effect turning the union of all those lists into one big list. Any other construction that would do the same would be just as good.)

I think that this example shows where our finite set intuition breaks when you switch to infinite sets.

If there were $n$ sets with $n$ elements each (with $n$ finite), you could use the same procedure, but much more likely you would go row-by-row or column-by-column.

Here you cannot do either because you can never enumerate the first row and switch to the second one (let alone third, fourth etc. - and similar for columns) so, somehow enumerating rows and columns "at the same time" is the only thing that can work.