How to prove $\sqrt{1+x} \le 1+\dfrac{x}{2}$, $x > -1$ using MVT
105268
prove inequality through mean value theorem
30385
Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
88425
Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
51653
Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
18016
Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
30688
Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.
108995
Prove $|e^{i\theta} -1| \leq |\theta|$
95293
Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
55582
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
73277
Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
122115
Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
126959
Prove $ \frac{1}{x+1} \le \ln(1+\frac{1}{x}) \le \frac{1}{x}$ with usage of the definition of a logarithmic function
138931
Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
109295
Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
146180
Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.