# Multiple choice question about the set $A = \{ m + n\sqrt{2} \}$. [duplicate]

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let $$A = \{m + n \sqrt{2}\}$$ where $$m,n$$ are integers, then

$$a.$$ $$A$$ is dense in $$R$$.

$$b$$. $$A$$ has no limit point in $$R$$.

$$c$$. $$A$$ has only countably many limit points in $$R$$.

$$d$$. only irrational numbers can be limit point of $$A$$

Set $$A$$ is similar to the set $$N \times N$$, so all points are isolated in $$A$$.
therefore option $$b$$ should be correct. In answer key, option $$a$$ is marked as correct. Please suggest if I am doing something wrong.

## marked as duplicate by Eevee Trainer, Lord Shark the Unknown, Saad, Paramanand Singh real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 27 '18 at 8:20

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• You say "Set $A$ can be represented as $(m,n\sqrt2),\ldots$". – Lord Shark the Unknown Dec 27 '18 at 5:43
• I edited the question! – Mathsaddict Dec 27 '18 at 5:46
• You now say "Set $A$ is similar to the set $\Bbb N\times\Bbb N$". Similar? How? – Lord Shark the Unknown Dec 27 '18 at 5:47

## 1 Answer

Consider $$\alpha=\sqrt2-1$$. Then $$\alpha^n\in A$$ for all $$n\in\Bbb N$$ (why?) and $$\alpha^n\to0$$. So $$0$$ is a limit point of $$A$$, which contradicts (d).

In general, if $$I$$ is an open interval in $$\Bbb R$$ then $$\alpha^n$$ is less than the length of $$I$$ for some $$n\in\Bbb N$$, and then $$m\alpha^n\in I$$ for some $$m\in \Bbb Z$$.

• Thanks, I got it. Can't this set be similar to $Q \times Q$ or $N \times N$ in terms of cardinality? – Mathsaddict Dec 27 '18 at 6:09