query: Prove: $\sup (A \cup B) = \max\{\sup(A),\sup(B) \}$ where $A,B \subseteq \mathbb{R}$
positive_passages.docid: 86505
positive_passages.text: Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$
negative_passages.docid: 89429
negative_passages.text: Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
negative_passages.docid: 135814
negative_passages.text: Prove certain subsets of $\mathbb R^n$ are subspaces
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 95935
negative_passages.text: Prove A = $(A \cap B) \cup (A \cap B^c)$ using the distributive and associative law.
negative_passages.docid: 143735
negative_passages.text: Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
negative_passages.docid: 127656
negative_passages.text: How to prove that if $A$ is infinite and $B$ is finite, then $|A\cup B|=|A|$?
negative_passages.docid: 123732
negative_passages.text: Proving that the maximum of two convex functions is also convex
negative_passages.docid: 88064
negative_passages.text: Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$
negative_passages.docid: 5132
negative_passages.text: Given $A\cap \overline{B}\neq \emptyset$, prove $A\cup B$ connected.
negative_passages.docid: 144360
negative_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 27899
negative_passages.text: Can we prove the existence of $A\cup B$ without the union axiom?
negative_passages.docid: 7112
negative_passages.text: Product of sets and supremum
negative_passages.docid: 90350
negative_passages.text: How can we replace $\sup$ with $\max$ in definition of subordinate norm for finite-dimensional vector space?
negative_passages.docid: 96848
negative_passages.text: Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$
negative_passages.docid: 127242
negative_passages.text: $|A\times B|= \text{max}(|A|,|B|)$ for infinite sets
negative_passages.docid: 106564
negative_passages.text: Suppose $A,B,C$ are sets. Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$.