Linked Questions

7
votes
4answers
2k views

$\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ dense in $\mathbb{R}$? [duplicate]

I'm guessing $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ is dense in $\mathbb{R}$. I'm having a mental block. How do you show that? (This is motivated by a different hypothesis: if $f$ is ...
1
vote
2answers
323 views

Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element $g_\epsilon\...
1
vote
3answers
404 views

Density in $\mathbb R$ [duplicate]

Let $A=\{a+b\sqrt{2}\mid a,b\in\mathbb Z\}$ and let $Y\subset A$ with $x\in Y$ iff $x\in [0,1]$. I'm trying to prove that $A$ is dense in $\mathbb R$ and already noticed that it is enough to show ...
4
votes
1answer
105 views

Are my arguments correct about limit points of $A=${$m+n\sqrt 2:m,n\in \mathbb Z$}? [duplicate]

Let $A=${$m+n\sqrt 2:m,n\in \mathbb Z$},then- $(1)A$ is dense in $\mathbb R$. $(2)A$ has only countable many limit points in $\mathbb R$. $(3)A$ has no limit points in $\mathbb R$. ...
1
vote
0answers
50 views

Limit Point of $A=\{{m+n\sqrt{2}}:m,n\in Z\}$ [duplicate]

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 $R$ ...
0
votes
1answer
50 views

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

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 ...
15
votes
3answers
5k views

For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$ [closed]

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way, $\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha\in\mathbb{Q}^c$ is a fixed irrational.
12
votes
2answers
873 views

If $\alpha$ is an irrational real number, why is $\alpha\mathbb{Z}+\mathbb{Z}$ dense in $\mathbb{R}$?

This is chapter $4$ exercise $25.b$ of Walter Rudin's Principles of Mathematical Analysis, this problem has occupied my mind for a long time, and I haven't been able to solve it, I would like to see ...
4
votes
5answers
4k views

How to show that $\{\sqrt m - \sqrt n : m,n \in \mathbb N\}$ is dense in $\mathbb R$? [duplicate]

Show that $S=\{\sqrt m - \sqrt n : m,n \in \mathbb N\}$ is dense in $\mathbb R$. I know the definition of dense as: A set S is dense in $\mathbb R$ if there exists $a,b \in \mathbb R$ such that $...
7
votes
1answer
323 views

The adherent values of $x_n=cos(n)$ are the interval $[-1,1]$

This question seems really hard, I'm trying to prove that the set of the adherent values of the sequence $x_n=\cos (n)$ is the closed interval $[-1,1]$, i.e., every point of this interval is a limit ...
2
votes
1answer
492 views

How do we know that $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ isn't closed?

The Minkowski sum of closed sets needn't be closed; $\mathbb{Z} + \sqrt{2}\mathbb{Z}$ is the canonical example. However, its not clear to me how to prove this. Question. How can we prove that $\...
2
votes
1answer
524 views

$\{m \alpha, m \in \mathbb Z\}$is dense in $[0,1]$ for $\alpha$ irrational

Let $\alpha$ is irrational and $S=\{\{n\alpha\}:n\in \mathbb{Z}\}$. I proved that for any positive integer $N$ $\exists m\in \mathbb{Z}$ such that $\{m\alpha\}<\frac{1}{N}.$ But how to use above ...
6
votes
1answer
619 views

How many limit points in $\{\sin(2^n)\}$? How many can there be in a general sequence?

Analysis question - given a sequence $\{a_n\}_{n=1}^\infty$, how many limit points can $\{a_n\}$ have? Initially I thought only $\aleph_0$, or countably many, because there are only countably many ...
6
votes
1answer
390 views

does there exist a discrete set whose image is dense

I want to know whether my proof is correct or not : Does there exist a descrete set whose image is dense in $S^1$ under the map $e^{2\pi ix}$ from $\mathbb{R}\rightarrow S^1$? my attempt is : We know ...
0
votes
1answer
201 views

Approximation of integer by multiple of irrational number

Obviously, for any $\epsilon >0$, there exist $m,n\in \mathbb{N}$ such that$$|\sqrt{2}-\frac{n}{m}|<\epsilon \; \textrm{.}$$ Is it also true that for all $\epsilon >0$, there exist $m,n\in \...

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