If $S=\left\{\frac{1}{p}+\frac{1}{q}\mid p, q\in \mathbb{N}\right\}$ then $\varlimsup S-\varliminf S=?$
My attempt since derived set of S is$\left\{\frac{1}{n}|n\in \mathbb{N}\right\}\cup \{0\}$ and so by taking difference of greatest and smallest limit point of above set we find $$\varlimsup S-\varliminf S=1$$
Am I correct?