Density of a set in $\mathbb R$

Question

Is the set $$K=\{\sqrt{m}-‎\sqrt{n} : m,n \in \mathbb{N}\}$$ dense in $\mathbb{R}$? It would be appreciated if someone can help me.

Answer 1

Take $u,v\in \mathbb{R}$ such that $u<v$. Since $\sqrt{m}-\sqrt{m-1}\to 0$ as $m\to +\infty$, there exists $m_0$ such that $0<\sqrt{m}-\sqrt{m-1} <v-u$ for $m\geq m_0$. Now take an $n_1$ such that $u+\sqrt{n_1}>\sqrt{m_0-1}$, and let $m_1$ be such that $\sqrt{m_1-1}\leq u+\sqrt{n_1}<\sqrt{m_1}$. We have $u<\sqrt{m_1}-\sqrt{n_1}$. Now, because $m_1\geq m_0$, we get $$\sqrt{m_1}-\sqrt{n_1}\leq \sqrt{m_1}-\sqrt{m_1-1}+\sqrt{m_1-1}-\sqrt{n_1}<v-u+u=v.$$

Hence we have shown that, for all $u$ and $v$ with $u<v$, there exists an element of $K$ in $]u,v[$, and so $K$ is dense in $\mathbb{R}$.