Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
6382
Showing $\frac{x}{1+x}0$ using the mean value theorem
94409
showing that the sequence $a_n=1+\frac{1}{2}+...+\frac{1}{n} - \log(n)$ converges
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
26962
Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$
83313
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
12020
Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
98768
Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
125680
Show that $(1+u) \log (1+u) - u \ge \frac{u^2}{2(1+u/3)} $
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
102738
Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction.
72600
Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
145274
Prove by induction $\sum \frac {1}{2^n} < 1$
117262
Proving a logarithm inequality
149889
Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
15104
Show that $\big|\frac{a-b}{1-\overline{a}b}\big|=1$, when either $|a|=1$ or $|b|=1$.
91000
Prove that $\frac{1}{a+1}0$