Prove that $|x_{n+1} - x_n| \le \theta |x_n - x_{n-1}|$ for some $0 < \theta < 1$
112430
Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$
3820
Prove that $||x|-|y||\le |x-y|$
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
147841
Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$
139383
Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
4315
show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
106585
Show $d_{1}(x_{n},x)\rightarrow{0}$ if and only if $d_{2}(x_{n},x)\rightarrow{0}$.
85217
Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$
104447
Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
107774
Prove $\|x\|_1\le \sqrt n \|x\|_2$
101286
How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
100375
Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
111316
PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
149056
Prove that for all $n\geqslant 1$ we have $F_n<{\left(\frac 74\right)}^n$.
122971
how to prove that $\ln(1+x)< x$
118613
Proving an inequality: $|1-e^{i\theta}|\le|\theta|$