Prove that: $\sqrt{2\sqrt{3\sqrt{4\cdots\sqrt{n}}}}<3,\,\forall n\in\mathbb N.$
21457
Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
52157
Prove that $d(n)\leq 2\sqrt{n}$
108588
Improving bound on $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$
25648
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
121694
How to prove $\log n \leq \sqrt n$ over natural numbers?
1401
$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation
87411
Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
66082
Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
49750
Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$
140409
Prove that $\sqrt{n} > \ln n$
34906
Showing $[\mathbb{Q}(\sqrt[4]{2},\sqrt{3}):\mathbb{Q}]=8$.
58284
How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?
98946
Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
28218
Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
153763
Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
142992
Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
46831
Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $