If $A_i$ is countably infinite set for $i=1$ to infinite then $\bigcup_{i=1}^{\infty} A_i$ is countably infinite.

Question

If the proof is done by a basic matrix diagonalwise model array then there arise a question that what will happen if 2 elements are equal? How can one make a one to one correspondence with the set of natural numbers?

Answer 1

First, suppose that the sets $A_i$ are pairwise disjoint. Let $f_i:\omega\rightarrow A_i$ onto and consider de function $f:\omega\times\omega\rightarrow \cup_i A_i$ suc that $f(n,m):=f_n(m)$. It's clear that $f$ is also onto, so $|\cup_iA_i|\leq\aleph_0$ (here we use AC).

In the general case, define $B_i=A_i\times\{i\}$. The function $g:\cup_i B\rightarrow \cup_i A_1$ given by $g(a,i)=a$ is also onto. So $\cup_iA_i$ must be infinite countable. (This paragrph says to you always can assume (WOLG) that the $A_i$ are disjoint).