Prove if $a>b>0$ and $n\in\mathbb{N}$ then $a^{1/n}-b^{1/n}<(a-b)^{1/n}$
1478
An inequality for powers of reals: $a^{1/n}-b^{1/n}<(a-b)^{1/n}$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
18693
If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
46831
Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
44897
Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$
55266
Prove that $(n+1)^{n-1}<n^n$
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
100375
Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
98496
Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
4315
show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
12954
Prove if $ac>bc$ and $c>0$, then $a>b$
118244
An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$
127293
Prove that $(a-b) \mid (a^n-b^n)$
81618
Prove $||a|-|b| | \leq |a-b|$
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
138212
Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$
81775
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$