query: If $\|\cdot\|_{1}\le\|\cdot\|_{2}$ then $\|\cdot\|_{2}\le M\|\cdot\|_{1}$
positive_passages.docid: 15839
positive_passages.text: equivalent norms in Banach spaces of infinite dimension
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 66707
negative_passages.text: How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $
negative_passages.docid: 62342
negative_passages.text: $x_1 \cdot \dots \cdot x_n=1 \implies x_1 + \dots + x_n \ge n$
negative_passages.docid: 10859
negative_passages.text: Prove $\sup(f+g) \le \sup f + \sup g$
negative_passages.docid: 85129
negative_passages.text: If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
negative_passages.docid: 80009
negative_passages.text: Prove that $| |a|-|b| | \le |a-b|$
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 6128
negative_passages.text: If $a \cdot a = 1$ then $a = 1$ or $a = -1$
negative_passages.docid: 104447
negative_passages.text: Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
negative_passages.docid: 98946
negative_passages.text: Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
negative_passages.docid: 144372
negative_passages.text: How to Prove that if $A$ is bounded, then $A \cdot A$ is bounded?
negative_passages.docid: 77777
negative_passages.text: If $0\le a,b,c\le 1$, then $\frac a{1+bc}+\frac b{1+ac}+\frac c{1+ab}\le 2$
negative_passages.docid: 122330
negative_passages.text: Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.
negative_passages.docid: 34929
negative_passages.text: Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.
negative_passages.docid: 151520
negative_passages.text: About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
negative_passages.docid: 107142
negative_passages.text: If $(x_n) \to x$ then $(\sqrt[n]{x_1x_2\cdots x_n}) \to x$