Let $A=\{{m+n\sqrt{2}}:m,n\in Z\}$, where $Z$ stands for the set of all integers .Then which of the following is correct

(a) $A$ is dense in $R$

(b) $A$ has only countably many limits points in $R$

(c) $A$ has no limit point in $R$

(d) only irrational numbers can be limit points of $A$

If m=0 and n=1 .Then $\sqrt{2}\in A$ and $(\sqrt{2}-0.5,\sqrt{2}+0.5)\cap A\sim{\sqrt{2}}=\Phi$ .

Then , only option c is correct. Please correct if I am wrong . I am not able to find any counter examples.

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  • Strong hint: You should add the tag [tag:group-theory] to your question. – Did Jul 24 '18 at 17:30
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    You are certainly wrong with your assertion that $(\sqrt{2}-0.5, \sqrt{2}+0.5)\cap A = \sqrt{2}$; for instance, $1 = 1+0\cdot\sqrt{2}$ is in that interval... – Steven Stadnicki Jul 24 '18 at 17:34

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