I saw a proof of the result using Zorn's Lemma:
Statement: If $X$ is a metric space in which every uncountable set has a limit point,then $X$ is separable.
The proof was a bit constructive and does not seem intuitive to me.So,I want to do it without Zorn's Lemma.I was proceeding by assuming $X$ is not separable.Then I was trying to find an uncountable set which has no limit point.But I got stuck and cannot find a way out.Can someone show me a way to prove the above theorem without using Zorn's Lemma.