Limit point of set $\{\sqrt{m}-\sqrt{n}:m,n\in \mathbb N\} $ How can I calculate the limit points of set $\{\sqrt{m}-\sqrt{n}\mid m,n\in \mathbb N\} $? 

The answer is $\mathbb{R}$, as we can see here, for $x\in (0,\infty)$ and $\epsilon >0$, there are $n_0 , N \in \mathbb{N}$ such that $\sqrt{n_0 +1}-\sqrt{n_0} <1/N<\epsilon /2$. Now we can divide $(0,\infty)$ to pieces of length $1/N$, so there is $k\in \mathbb{N}$ such that $k(\sqrt{n_0 +1}-\sqrt{n_0})\in N_{\epsilon} (x)$.
The proof for $(-\infty , 0)$
is the same.
