query: Prove that $\sqrt{2\sqrt{3\sqrt{...\sqrt{(n - 1)\sqrt{n}}}}} < 3$.
positive_passages.docid: 21457
positive_passages.text: Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
negative_passages.docid: 62058
negative_passages.text: How to prove $1$,$\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent over $\mathbb{Q}$?
negative_passages.docid: 27147
negative_passages.text: Proving $n! > n^3$ for all $n > a$
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 51730
negative_passages.text: How to prove this inequality $3^{n}\geq n^{2}$ for $n\geq 1$ with mathematical induction?
negative_passages.docid: 64457
negative_passages.text: Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 34491
negative_passages.text: Is there a way to show that $\sqrt{p_{n}} < n$?
negative_passages.docid: 42976
negative_passages.text: How do you prove the follwing $|\sqrt[n]{x}-\sqrt[n]{y}| \le \sqrt[n]{|x-y|}$
negative_passages.docid: 87682
negative_passages.text: Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$
negative_passages.docid: 139718
negative_passages.text: Prove $6 \nmid [\left( \sqrt[3]{28} - 3 \right)^{-n}]$
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: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 140409
negative_passages.text: Prove that $\sqrt{n} > \ln n$
negative_passages.docid: 99052
negative_passages.text: Prove that $( 1 + n^{-2}) ^n \to 1$.
negative_passages.docid: 90276
negative_passages.text: Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
negative_passages.docid: 129171
negative_passages.text: Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$