Let $E$ be a normal vector space, $A$, $B\subset E$ closed, nonempty subsets.

We denote by $c = \inf\{\|a − b\| \mid a\in A, b\in B\}$.

I gotta find $2$ closed sets $A$, $B$ such that $c = 0$ and $A\cap B = \varnothing$.

And then, for these sets, give an example of sequences $\{a\_{n}\}$ and $\{b\_{n}\}$ such that $\|a\_{n} - b\_{n}\| \to 0$.

I have the idea of $A = N - \{0\}$ and $B = \{(n+1)/n \mid n \in\mathbb{N}\_{>=2}\}$ for my closed sets.

Is it okay? I'm kinda lost about the sequences

Thank You for your help