If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)
I'm struggling with this problem for a week.
I'm talking about this in Metric space.
Here's the part of argument in Rudin PMA p.45; Let $X$ be limit point compact. Fix $\epsilon>0$ and $x_0 \in X$. Having chosen $x_0,\ldots,x_j \in X$, choose $x_{j+1}$, if possible, so that $d(x_i,x_{j+1})≧\epsilon$ for $i=0,\ldots,j$. Then this process must stop after a finite number of steps.
Here, Dependent Choice is used.
I know that ‘Limit point property $\Rightarrow$ separable’ is unprovable in ZF. (You can see this: If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF))
I want to prove this in ZF+AC$_\omega$. Help.
So far, I proved that [Limit point compact $\Rightarrow$ separable] $\Rightarrow$ [Limit point compact $\Rightarrow$ Compact].
I need this to complete my proof.