Lindelöf and separable metric space

Question

Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?

Answer 1

Hint: for each $n \in \mathbb{N}$, the cover $A_n := \{B_{\frac{1}{n}}(x) : x \in X\}$ of $X$ has a countable subcover, $$ \mathcal{B}_n = \{B_{\frac{1}{n}}(x^{(n)}_j)\}_{j \geq 1} \subseteq A_n. $$

In particular, each point of $x$ is in some ball of $B_n$. That is, each point of $x$ is at distance less than $\frac{1}{n}$ from some $x_j^{(n)}$. What does this tell you about $\bigcup_{n \in \mathbb{N}} \{x^{(n)}_j\}_{j \geq 1}$?