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Which of the following is true about set $A$?

$A=\{m+n \cdot 2^{1/2} : m, n \text{ are integers}\}$

  1. $A$ is dense in $\mathbb R$.
  2. $A$ has only countable number of limit point in $\mathbb R$.
  3. $A$ has no limit point in $\mathbb R$.
  4. Only irrational number is limit point of $A$.

please explain.

Dave L. Renfro
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1 Answers1

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  1. is true. Implying the others are false.

For $x\in \Bbb R$ let {$x$} be the fractional part of $x.$ That is, if $z\in \Bbb Z$ and $z\le x<z+1$ then {$x$}$=x-z.$ Let $F(x)=\{ n${$x$}$: n\in \Bbb N\}. $ If $x$ is irrational then $F(x)$ is dense in $[0,1].$ There are many ways to prove this. I recommend the link in the comment to the Q by @Kavi Rama Murthy.

DanielWainfleet
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  • I don't disagree with your proof, but I'm genuinely curious about why 2 is false. Isn't A itself countable? – harwiltz Oct 15 '20 at 14:34
  • @harwiltz . $A$ is dense in $\Bbb R$ so every $r\in\Bbb R$ is a limit point of $A$...... $\Bbb Q$ is another example of a countable dense subset of $\Bbb R$ – DanielWainfleet Oct 15 '20 at 14:42
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    thanks a lot. That's pretty neat. I also interpreted 2 as there are countably many limit points of R contained in A. – harwiltz Oct 15 '20 at 20:16