(a) Let $A_{n}=B_{1/n}((0,0))$ in $\mathbb{R}^{2}$ with the usual metric. Show that $\bigcap_{n=1}^{\infty}A_{n}$ is not open.
(b) Find an infinite collection of distinct open sets in $\mathbb{R}^{2}$ with the usual metric whose intersection is a nonempty open set.
Attempt at (a): Clearly $(0,0)\in A_{n}$, $\forall n\in\mathbb{N}$ so $0\in\bigcap_{n=1}^{\infty}A_{n}$. Let $(a,b)\not=(0,0)$ s.t. $(a,b)\in\bigcap_{n=1}^{\infty}A_{n}$.
Since $d((a,b),(0,0))= \sqrt{a^2+b^2}\in\mathbb{R^+}>0$, can we say that by the archimidean property of $\mathbb{R}$ there exists an $nā\mathbb{N}$ st. $\frac{1}{n}\lt d((a,b),(0,0))$ which means $(a,b)=(0,0)$ or $(a,b)\notin A_n$?
Attempt at (b): I need a collection of sets ${A_1,A_2,...}$ s.t. $\bigcap^{\infty}_{n=1} A_n$ is open. I'm using to finding the opposite -- an infinite collection of open sets whose intersection is closed as in (a). I'm having difficulty finding an open set $A$ s.t. $A\subseteq A_n$ $,\forall n$.
Edit: Had an idea for (b): The set $A_n=B_{1+n}((0,0)) \forall n\in\mathbb{N}$. The infinite intersection should be the open set $(-1,1)$, correct?