Are $\sqrt{m}-\sqrt{n}$ Dense In $\mathbb{R}$ ?

Consider an interval $(a,b)$ and assume, without loss of generality, that $a>0$ . There exists a positive integer $p$ such that $d=\sqrt{p+1}-\sqrt{p}=\frac{1}{\sqrt{p+1}+\sqrt{p}}<b-a$ . Now there exists a positive integer $q$ such that $qd=\sqrt{q^2(p+1)}-\sqrt{q^2p}$ is on $(a,b)$ , as required.

This is equivalent to construct, for every $a\ge 0$ , a sequence $(m_k,n_k)$ such that $\sqrt{m_k}-\sqrt{n_k} \to a$ . To make it easier, let's look for sequences with $m_k = k$ . We guess that for every $k$ , we can choose $n_k$ to be the largest integer $n$ for which $a \le \sqrt{k}-\sqrt{n}$ , that is, $n_k =$ integer part of $(\sqrt{k}-a)^2$ . But this is not difficult to prove.