In $\mathbb{R^2}$ with the usual metric, consider $C_n=\{(x,y)\in\mathbb{R^2} : x^2 + y^2 = 1/n^2\}$, that is, the circumference centered at the origin with radius $1/n$. Now consider the union of all those, that is, $C=\cup_{n\in\mathbb{N}}C_n$
In an exercise, I'm asked to calculate the closure, interior and boundary. Now, the intuitive solution to me is that the closure is $C\cup\{(0,0)\}$, the boundary is the same, and the interior is empty. But how to prove this formally?
EDIT: Regarding what I've tried, we firstly know that $C$ is in the closure. Then, do we have anything else? By using the characterization of the closure by sequences, its easy to prove that you can find a sequence that converges to the origin, hence it also is in the closure. Now, what I do is consider a point $p\notin C, p\neq(0,0)$, and try to prove that there are no possible sequences in $C$ that converge to $p$. I've also tried to prove that there exist an open balls centered at that point contained in $X-C$. The problem for me always arises at the time of formalizing those thoughts.