Prove that every infinite set has a countable dense subset.
The Attempt at a Solution
I have almost no idea how to solve this problem of my analysis homework. I was thinking that i need to show that there is a countable subset that has all points of the initial set as limit points. So i was thinking of saying that that any open infinite set can be written as a countable union of subsets or like neighbourhoods, and using something like a pigeon hold principle to say that each of these neighbourhoods must contain some points in the countable subset. but we have countable nbhds and countable points, i dont think the pigeonhole principle works here... I'm probably way off the solution... ANY HELP?