Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 8E / a

Write the sets as $A_1,A_2,\dots$, and list each nonempty countable set as $A_i=\{a_{i1},a_{i2},\dots \}$. Enumerate the pairs $(i,j)\in {\mathbb{N}}^2$ along successive diagonals, as in the Cantor pairing function, and output $a_{ij}$ when it has not appeared before. Every member of ${\bigcup }_i{A}_i$ eventually appears, proving that a countable union of countable sets is countable.