In $(\mathbb R^2, ||\cdot||_{\infty})$, let:
$A_0 = ]0,1] \times \{0\}$
$A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$.
$A = \cup_{n=0}^{\infty} A_n$
It is required to prove that:
$\bar A = A \cup [(0,0), (0,1)]$
I proved that $A \cup [(0,0), (0,1)]$ is contained in $\bar A$.
For the other inclusion, I should probably take a sequence in $A$ and prove that its limit is either in $A$ or in that segment. I did a proof on scratch, but it lacks rigour, and after many attempts, I had no luck constructing a rigorous proof. Hence I am asking for a rigorous proof for that, not a hint, because I already know what is going on.
Though, any other suitable approaches are appreciated.
Thank you.