Given are $l_2$, the collection of all real sequences for which $\sum_{n =1}^\infty |x_n|^2 < \infty$ and subset $A = \{x\in l_2: |x_n|\leq \frac{1}{n}, n = 1,2,...\}$.
I need to show that A is closed. What I've tried to do is this:
Pick a point $x \in l_2\backslash A$; there exists a $\delta$ such that $B_\delta(x) \subset l_2\backslash A \Rightarrow l_2\backslash A$ is open $\Rightarrow A$ is closed.
Question: Is my solution elaborate enough? I have just been introduced to the concept of $l_2$ and I'm not sure whether you can take balls around points as I have done above.
Thanks in advance!