Proof of set closer and more

Question

So this is a topological question I got on my exam some time ago " For $A \subset X$ and $ \epsilon \gt 0 $ let $$B(A, \epsilon) = {x \in X : \underset{a \in A}{\exists}d(x,a)\lt \epsilon}$$ -Prove thet $B(A,\epsilon)$ is an open set

-Prove then $\overline{A} = \bigcap^\infty_{n=1} B(A, \frac{1}{n})$"

Obviously I wasn't able to do it, hence I'm posing it here in hope thet someone will shed some light on this question

Answer 1

Hint for first part: If $x \in B(A,\epsilon)$ then there exists $a \in A$ such that $d(x,a)<\epsilon$. Show that $B(x, \frac {\epsilon -d(x,a)} 2) \subseteq B(A,\epsilon)$.

Hint for second part: $\overline {A}=\{x: d(x,A)=0\}=\cap_n \{x: d(x,A) <\frac 1 n\}$.