Help to show relation between ${\sqrt {x + 1}} - {\sqrt x} \leq {\frac{1}{2\sqrt x}}$
12020
Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
145884
Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
55582
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
44233
How to prove this inequality $ x + \frac{1}{x} \geq 2 $
103030
Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$
136297
Evaluate $\lim_{x \to 1} \frac{\sqrt[3]{x} -1}{\sqrt{x} -1}$
108782
Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$
104272
Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$
26606
Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$
14952
Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
105047
Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
118244
An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$
77130
How to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|<2\sqrt{|x-y|}$?
61782
How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
137791
Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
15205
Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$