Finding limit points of {$\sqrt n$}

Question

How can I find limit points of {$\sqrt n$}, where {.} represents the fractional part of a number. Intuitively it should be $[0,1]$, but what is a rigorous argument?

Answer 1

Here is a hint: $$\sqrt{n+1}-\sqrt{n}\to0.$$ These are the gaps in the sequence $\{\sqrt{n}\}$, with a few exceptions (the execptions being when $n+1$ is a square).

So given $a\in(0,1)$, look for those $n$ where $\{\sqrt{n}\}\le a<\{\sqrt{n+1}\}$, which is to say $\sqrt{n}\le k+a<\sqrt{n+1}$ for some integer $k$. I.e., $n\le (k+a)^2<n+1$. There are infinitely many such $n$, one for each $k$.