Hey guys, could you please check over my solution and let me know if I made a mistake somewhere?

Problem: Let be countable sets. Then show that the Cartesian product is uncountable. In other words, show that the set of countably infinite sets whose elements are natural numbers is uncountably large.

Approach: I know that this is solvable using a diagonalization argument. But I wanted to try something else. I attempted to surject this to real numbers.

Consider a subset of , say where holds all such elements of A where .

So I defined a function where:

For any element of B, sends that set to

This function is surjective because for all real numbers whose digits are: , we can find an element

The existence of this surjection implies that . And we also know that , so that . By transitivity, which means that must be uncountable.