Find $\inf\{\operatorname{Frac}(\sqrt{5}n) : n\in\Bbb N_+\}$ Find $\inf \{\operatorname{Frac}(\sqrt{5}n) \colon n \in \mathbb{N}_+\}$ 
Just for some information that $\operatorname{Frac}(x) \in [0,1)$ s.t $x - \operatorname{Frac}(x) \in \mathbb{Z}$ 
It looks to me that the $\inf$ is 0. I am not so sure how to prove it. 
 A: There is a Theorem by Dirichlet which says that the fract part of $n\alpha$ is dense in $[0,1)$ if $\alpha$ is irrational.
Anyhow, this can be proven by the Pigeonhole Principle, and the following basic idea is enough for what you want:
Divide $[0,1)$ in $m$ intervals of length $\frac{1}{m}$. Look at the fractional part of 
$\sqrt{5}, 2 \sqrt{5},.., (m+1) \sqrt{5}$. Two of them must be in the same mini interval, and then the fractional part of the difference must be in $(0, \frac{1}{m})$.
Done. 
A: Hint: Let $a_n$ be the fractional part of $\sqrt{5}n$, and let $N$ be a large positive integer. Consider $a_1, a_2, a_3, \dots, a_{N+1}$. Then by the pigeonhole principle there are $i$ and $j$, with $1\le i\lt j \le N+1$, such that $|a_j-a_i|\lt \dfrac{1}{N}$.  
This will not quite do it. But something fairly close will. 
A: Consider the line $L$ given by $y=mx$ on the two-dimensional real torus $T=([0,1]\times[0,1])/\sim$.  If $m$ is irrational, it is known that $L$ is dense in $T$.  In particular, $L$ intersects to a dense subset in $(\{0\}\times[0,1])\subset T$, which represents the points $mn$ for $n\in\mathbb{N}^+$.
