Hello everyone. I would greatly appreciate if someone could validate whether or not the following solution is correct.

Is it possible for a countably infinte set to have an uncountable collection of subsets such that the intersection of any two is finite?Problem:

Yes it is.Answer:

Proof:

Define to be a rational sequence converging to and and let . It is clear that if that is finite, otherwise both sequences would converge to the same value. Particularly this implies that thus . Lastly noting that we may conclude that is an uncountable collection of subsets of a countable set such that the intersection of any two is finite.

What do you think?