query: Prove that the supremum of the set $A = \{x \in \Bbb Q \; | x^2 < 2\}$ is $\sqrt{2}$ in $\Bbb R$
positive_passages.docid: 22224
positive_passages.text: Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.
negative_passages.docid: 47820
negative_passages.text: How can I show a supremum of a set is also its limit point?
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
negative_passages.docid: 143105
negative_passages.text: Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
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: 81203
negative_passages.text: Supremum and infimum of sets
negative_passages.docid: 98577
negative_passages.text: Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
negative_passages.docid: 96848
negative_passages.text: Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 88064
negative_passages.text: Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$
negative_passages.docid: 63526
negative_passages.text: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
negative_passages.docid: 56818
negative_passages.text: Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
negative_passages.docid: 10859
negative_passages.text: Prove $\sup(f+g) \le \sup f + \sup g$
negative_passages.docid: 68781
negative_passages.text: proving if $a \le b$ then $\sup(A) \le \inf(B)$
negative_passages.docid: 59424
negative_passages.text: Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
negative_passages.docid: 136403
negative_passages.text: Prove that $\sqrt{a}\leq\frac{1+a}{2}$