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Let $x_n \in[0,1)$ s.t $x_n - \sqrt{n}\in \mathbb{Z}$. Prove that $X=\{x_n | n\in \mathbb{N}\}$ is dense in $[0,1]$.

It it suffices to prove that $\overline{X}=[0,1].$ I was trying to use the fact that $p+q\sqrt{2}$ $(p,q \in \mathbb{N})$ is dense in $\mathbb{R}$. So I can choose $n=2q^2$ and $x_n=p+q\sqrt{2}$ close enought of $1$. Hence, $x_n=p+q\sqrt{2} - q\sqrt{2} = p \in \mathbb{Z}$.

I am pretty sure that this argument is invalid but I feel that the idea can be useful somehow.

yolomorphism
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  • The object of the exercise appears to begin with an arbitrary sequence $x_n$ with the assumed property holding for all $n$ and then show the desired result, not to construct $x_n$ (this may be an issue of exposition; it isn't clear if your attempt is trying to do construct a particular sequence, or somehow use the set you identify as a guide to working with an arbitrarily given $x_n$ satisfying the hypotheses). – leslie townes Feb 22 '21 at 01:49
  • In essence, you are being asked to show that, as $n$ runs through the positive integers, the fractional part of $\sqrt n$ is dense in $[0,1]$. Try to prove that, or try to see whether that question has already been asked and answered on this website before. – Gerry Myerson Feb 22 '21 at 02:03
  • https://math.stackexchange.com/questions/3423465/everywhere-dense-set-sqrtn-mod-1 – angryavian Feb 22 '21 at 02:04
  • Also https://math.stackexchange.com/questions/2963773/dense-set-of-numbers and https://math.stackexchange.com/questions/3185456/fractional-parts-of-square-roots-of-primes – Gerry Myerson Feb 22 '21 at 02:06

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$p+q\sqrt{2}$ with $p,q \in \mathbb{N}$ is not dense in $\mathbb{R}$, you probably mean $p,q \in \mathbb{Z}$. That's true, and an easy consequence of the irrationality of $\sqrt{2}$, but certainly not easier than a direct proof of your claim that the fractional part of $\sqrt{n}$ is dense in $[0,1]$: when $n$ grows from $m^2$ to $(m+1)^2-1$, the fractional part of the square root grows from $0$ to almost $1$, with differences $$\sqrt{n+1}-\sqrt{n}=\frac1{\sqrt{n+1}+\sqrt{n}}<\frac1{2m}.$$

NoNames
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