In the book Principles of Mathematical Analysis (1976) by Walter Rudin specifically, chapter $2$ theorem $2.12$, the theorem states the following :
Theorem 2.12 : Let $\{E_{n}\}$, $n=1,2,3…$, be a sequence of countable sets , and put : $$ S=\bigcup_{n=1}^{\infty}E_{n} $$ Then
$S$ is countable.
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Q1) What is the difference between the union of countable sets and the union of a sequence of countable sets? $$
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Q2) I did not quite understand the strategy of the proof Rudin presented due to the fact that I am trying to study analysis by myself which might have been a bad idea. $$
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On a side note, I hope I am not the only person who finds the notation used to represent a sequence to be inappropriate.
Best Answer
The strategy is to list all elements while making sure that 'all bases are covered'.
You organize and layout the elements on paper,
Note that while there is no bottom or right edge for this 'matrix picture', you can 'cross stuff out' by traversing from the left edge to the top edge using diagonals.
The elements in set $E_n$ are listed out in row $n$, with the first subscript $n$ of $x_{(n,k)}$ used for the set and the second subscript $k$ used for the enumeration of that set.
Then as Rudin crosses out the diagonals all elements will be crossed out. Notice that the third diagonal,
$\quad x_{(3,1)},x_{(2,2)},x_{(1,3)}$
'takes out' the entries where the subscripts for $x_{(n,k)}$ add up to $4$,
$\quad n + k = 4$
The $t^{th}$ diagonal takes out the $n + k = t + 1$ tabularly arranged elements.