query: Prove: $A,B\in\mathscr P(\Bbb R)\setminus\emptyset ,a \lt b\ \forall a \in A,\forall b \in B\implies \sup A \lt \inf B$
positive_passages.docid: 147941
positive_passages.text: comparing sup and inf of two sets
negative_passages.docid: 139841
negative_passages.text: Proving $A\setminus(B\setminus C)$ is not equal to $(A\setminus B)\setminus C$
negative_passages.docid: 102168
negative_passages.text: Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
negative_passages.docid: 48783
negative_passages.text: Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
negative_passages.docid: 27170
negative_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 109682
negative_passages.text: Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
negative_passages.docid: 151852
negative_passages.text: How to prove $A \subset B \iff A \cup B = B$
negative_passages.docid: 67709
negative_passages.text: Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$
negative_passages.docid: 52135
negative_passages.text: Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
negative_passages.docid: 43462
negative_passages.text: Disprove $\mathcal{P}(A\Delta B) \setminus \{\emptyset\} = \mathcal{P}(A) \Delta\mathcal{P}(B) $. Is my proof good?
negative_passages.docid: 64320
negative_passages.text: Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
negative_passages.docid: 122326
negative_passages.text: Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 7112
negative_passages.text: Product of sets and supremum
negative_passages.docid: 152817
negative_passages.text: Proving $\sup B=\inf A$
negative_passages.docid: 11817
negative_passages.text: Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
negative_passages.docid: 80255
negative_passages.text: How to prove that $|a-b|<\epsilon$ implies $|b|-\epsilon<|a|<|b|+\epsilon$?