Consider the set $\{\frac{1}{n} + \frac{1}{m}: n,m \in \mathbb{N} \}$. Is this set dense in some interval of $\mathbb{R}$?

More generally let $S_k= \{\sum_{i=1}^k \frac{1}{n_i}: n_i \in \mathbb{N} \} $. Is this set dense in some interval of $\mathbb{R}$?

I think the answer to both questions is no, but I was not able to come up with a proof. I am a physics student so I apologise if this question is naive for this website.