Distance between a real number and a whole number Prove that if $a$ is real and $n$ natural. The distance between one of the numbers $a,2a,3a,...,na$ and a whole number is  at most $\frac{1}{n}$.
This is a problem from discrete math, but hints from analysis would be appreciated.
 A: Any real $a$ can be written as $a=\lfloor a\rfloor+\{a\}$, where $\lfloor \cdot\rfloor$ is the greatest integer function and $\{a\}$ is the fraction part of $a$. Now note that $\{a\}\in[0,1)$ for any $a\in\mathbb{R}$. Partition the interval $[0,1)$ into $A_1,\cdots,A_n$ where $A_k=\left[\frac{k-1}{n},\frac{k}{n}\right)$ for $k\in\{1,\cdots,n\}$. Now $\{a\}$ must belong to one of these subintervals, say $A_i$. Then $\{xa\}\in\left[0,\frac{xi~mod~n}{n}\right)$. Now $\mathbb{Z}_n^*$, the nonzero elements of integers modulo $n$, is a multiplicative group. Hence for every integer $i\leq n,\exists~ x\leq n$ such that $xi\equiv1\mod n$. That is for some $x\leq n$, $xa$ is within a distance of $\frac{1}{n}$ from the integer $\lfloor x a\rfloor$ .   
A: For the analysis intuition:
This is actually an interesting question of looking at the circle - $S^1$ and an angle $\theta$ and asking to prove that one of $\theta,\ ...,\ n\theta$ is close to $0$ by $\frac{1}{n}$.
To understand what I said above just look at everything mod $1$. I think now you can visualise why the statement is correct.
For the proof:
To prove that just divide to cases: 


*

*$\theta \in [0,\frac{2\pi}{n}]$.

*$\theta \in [\frac{2\pi}{n},2\frac{2\pi}{n}]$.


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k. $\theta \in [(k-1)\frac{2\pi}{n},k\frac{2\pi}{n}]$.
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n. $\theta \in [(n-1)\frac{2\pi}{n},n\frac{2\pi}{n}]$.
And from here it's quite easy.
If I'm wrong feel free to brutally scold me.
