query: Define $-S = \{-x \mid x \in S\}$ , prove $\sup(-S) = -\inf(S)$ and $\inf(-S) = - \sup(S)$, with $S$ is bounded both sides
positive_passages.docid: 1521
positive_passages.text: Supremum and Infimum
negative_passages.docid: 135694
negative_passages.text: Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
negative_passages.docid: 97772
negative_passages.text: An example of a function which satisfies $\sup_{y}\inf_{x}f(x,y)<\inf_{x}\sup_{y}f(x,y)$
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 22224
negative_passages.text: Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.
negative_passages.docid: 30770
negative_passages.text: Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
negative_passages.docid: 102478
negative_passages.text: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
negative_passages.docid: 88278
negative_passages.text: Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
negative_passages.docid: 8414
negative_passages.text: Proof verification: Set of partial limits of a sequence is closed
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 144372
negative_passages.text: How to Prove that if $A$ is bounded, then $A \cdot A$ is bounded?
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 14713
negative_passages.text: $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
negative_passages.docid: 80421
negative_passages.text: Show that $a - b \mid f(a) - f(b)$
negative_passages.docid: 117257
negative_passages.text: Proving the supremum of the set {$\cos (1/x)$} is 1