query: Show that $\alpha+ \beta$ is the supremum of $A + B$.
positive_passages.docid: 24856
positive_passages.text: Proving sup(A + B) = sup A + sup B
negative_passages.docid: 76140
negative_passages.text: The supremum norm is submultiplicative
negative_passages.docid: 58629
negative_passages.text: Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.
negative_passages.docid: 46831
negative_passages.text: Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
negative_passages.docid: 115176
negative_passages.text: Can we say that $\left(a+b\right)^{\alpha}>a^{\alpha}+b^{\alpha}$ for all $a,b>0$ and $\alpha>1$?
negative_passages.docid: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
negative_passages.docid: 76863
negative_passages.text: How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
negative_passages.docid: 133382
negative_passages.text: Proof that $a+b+\frac{1}{1+\sqrt{ab}}\geq\frac{5}{2}$
negative_passages.docid: 43246
negative_passages.text: Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
negative_passages.docid: 67483
negative_passages.text: Show that $|a-b|<\epsilon \Rightarrow |a|<|b|+\epsilon$
negative_passages.docid: 87536
negative_passages.text: Supremum of the infimum
negative_passages.docid: 136789
negative_passages.text: How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 53857
negative_passages.text: Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
negative_passages.docid: 33030
negative_passages.text: Proof that $\inf A = -\sup(-A)$
negative_passages.docid: 96217
negative_passages.text: If a and b are non-negative real numbers then demonstrate inequality
negative_passages.docid: 79491
negative_passages.text: Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$