query: Is the set {$ 1,\frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8},\frac{1}{16},\frac{3}{16},....$} dense in [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: 104274
negative_passages.text: Express $C$ interms of the sets $A_n$
negative_passages.docid: 27253
negative_passages.text: Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$
negative_passages.docid: 107564
negative_passages.text: If two sets have a natural (asymptotic) density, does their union?
negative_passages.docid: 118400
negative_passages.text: Is there a monotonic function discontinuous over some dense set?
negative_passages.docid: 153072
negative_passages.text: Guessing a subset of $\{1,...,N\}$
negative_passages.docid: 132068
negative_passages.text: Write the set $\{1, 6, 11, 16,\ldots\}$ in compact (concise) notation.
negative_passages.docid: 130418
negative_passages.text: Is ${\{\emptyset,\{\emptyset\},\{\{\emptyset\}\},\{\{\{\emptyset\}\}\},...\}} $ a set?
negative_passages.docid: 15695
negative_passages.text: Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
negative_passages.docid: 107234
negative_passages.text: Is the mean of equal length subsets always equal to the mean of the set?
negative_passages.docid: 124060
negative_passages.text: Are these subsets of the powerset P(N) countable?
negative_passages.docid: 41408
negative_passages.text: Which of the following sets are compact:
negative_passages.docid: 132382
negative_passages.text: Showing a set is nowhere dense in $C([0,1])$
negative_passages.docid: 64941
negative_passages.text: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
negative_passages.docid: 96931
negative_passages.text: Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
negative_passages.docid: 45216
negative_passages.text: Cardinality of $[0,1]$
negative_passages.docid: 18530
negative_passages.text: Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?