query: How to show that $\bigcup_{n \in \mathbb{N}} \{ \frac{k}{2^n} : 1 \leq k \leq 2^n \}$ is dense in $\mathbb{Q} \cap [0,1]$?
positive_passages.docid: 98755
positive_passages.text: How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?
negative_passages.docid: 107370
negative_passages.text: An open covering of $\mathbb{Q} \cap [0,1]$ that does not contain any finite subcovering
negative_passages.docid: 68490
negative_passages.text: For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
negative_passages.docid: 6523
negative_passages.text: Prove that the space has a countable dense subset
negative_passages.docid: 103314
negative_passages.text: Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty?
negative_passages.docid: 97064
negative_passages.text: Dense set in an infinite cofinite space
negative_passages.docid: 95294
negative_passages.text: Dense subset of $C[0,1]$ which preserves rationals
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 158582
negative_passages.text: Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
negative_passages.docid: 50934
negative_passages.text: Let $x_n \in[0,1)$ s.t $x_n - \sqrt{n}\in \mathbb{Z}$. Prove that $X=\{x_n | n\in \mathbb{N}\}$ is dense in $[0,1]$.
negative_passages.docid: 158018
negative_passages.text: Everywhere Super Dense Subset of $\mathbb{R}$
negative_passages.docid: 4381
negative_passages.text: Proof that the set of irrational numbers is dense in reals
negative_passages.docid: 111898
negative_passages.text: Is $\{ (e^{2\pi i a n},e^{2\pi i b n }) : n \in \mathbb{Z} \}$ dense in the torus, where $a,b$ irrationals such that $a/b$ irrational?
negative_passages.docid: 36546
negative_passages.text: For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$
negative_passages.docid: 44430
negative_passages.text: Is there an open dense set $S \subset [0,1]$ such that $m(S)<1$?
negative_passages.docid: 126821
negative_passages.text: Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
negative_passages.docid: 145726
negative_passages.text: For a subfield $K$ of $\Bbb C$ with $K\nsubseteq \Bbb R$, show $K$ is dense in $\Bbb C$.