# Linked Questions

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### $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ 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 ...
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1answer
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### 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 ...
1answer
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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 $\... 1answer 524 views ###$\{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}.$But how to use above ... 1answer 619 views ### How many limit points in$\{\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$, or countably many, because there are only countably many ... 1answer 390 views ### does there exist a discrete set whose image is dense I want to know whether my proof is correct or not : Does there exist a descrete set whose image is dense in$S^1$under the map$e^{2\pi ix}$from$\mathbb{R}\rightarrow S^1$? my attempt is : We know ... 1answer 201 views ### Approximation of integer by multiple of irrational number Obviously, for any$\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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