Dense set of numbers

Show that the set $\{\sqrt{n}-[ \sqrt{n}]; n\in \Bbb N\}$ is dense in $\Bbb [0,1]$.

I showed that for every real number $a$ such that $\ 0<a<1 $ there is at least one number $x$ from the set such that $\ a< x$.

And for every real number $b$ such that $\ 0<b<1 $ there is at least one number $y$ from the set such that $\ b>y $

But I don't know how to prove that: for every real number $a$ such and every real number $b$ such that $\ 0<a<b<1$ there is at least one number $z$ from the set such that $\ a<z<b $