Proving that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

Question

I need some help with this problem:

Let be $(X,\rho)$ a metric space, $A \subseteq X$ and $b \in X$. The distance from $A$ to $b$ is defined as $\rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}$. Prove that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

Proof: Let be $b \in \overline{A}$, then for all $r > 0$ we have that $B_r\,(b) \cap A \neq \emptyset$...

I'm sutck here, because I don't know hot keep going on my demonstration.

I appreciate all your help, thanks!

Answer 1

Using the idea that you have started you may want to want to choose $r=\frac{1}{n}$. So when you say $b \in\bar{A}$, then you can say for each $n \geq 1$, $B_{\frac{1}{n}}(b) \cap A \neq \phi$. Now for each $n$ you have a point $a_n \in A$ such that $a_n \in B_{\frac{1}{n}}(b) \cap A$. So $\rho(b,a_n) \leq \frac{1}{n}$.

Now you may conclude that $\rho(b,A) \leq \frac{1}{n}$ for all $n \geq 1$. Furthermore complete the argument by showing that infimum will be $0$.