Show that if $1> x>0$, then $x-1 ≥ \ln(x) ≥ 1−\frac{1}{x}$
43226
Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$
101701
Prove that $\ln x$ is not uniformly continuous on $(0,1]$
100234
Proving that $x^n$ converges to $0$ whenever $|x| < 1$
13902
Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
24683
For i.i.d. $\{X_n\}$ proving $\frac1n\max\limits_{1\leqslant k\leqslant n}|X_k|\xrightarrow{\mathrm{a.s.}}0\Leftrightarrow E(|X_1|)<+∞$
19159
Convergence of $\frac{1}{(\ln n)^{\ln n}}$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
25775
How to prove $(1+1/x)^x$ is increasing when $x>0$?
68990
Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$
82961
Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
6100
Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
9746
Determine the following limit as x approaches 0: $\frac{\ln(1+x)}x$
15809
The problem of x = ln(x)
127397
$z\overline{w}\ne 1 $ Prove that $|\frac{w-z}{1-\overline{w}z}|\le 1|$ if $|z|\le 1$ and $|w|\le 1 $
103757
how can I prove that $\frac{\arctan x}{x }< 1$?
45742
Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
60321
Use the definition of derivative to prove $\lim_{x\to 0}\ln(x+1)/x =1$