For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$
68490
For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
132382
Showing a set is nowhere dense in $C([0,1])$
30568
Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$
30478
How to prove that a number is irrational
40524
Let $A \subseteq (0,\infty$) such that $\inf A>0$. Let $\frac{1}{A}=\{\frac{1}{a},a\in A\}$. Prove that $\sup \frac{1}{A}=\frac{1}{\inf A}$.
97968
Does $\big|\sum_{k=1}^na_k\big|\leq\sqrt{n}$ imply the convergence of $\sum_{1}^\infty \frac{a_k}{k}$?
26109
Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
15344
Let $a_n=\cos(a_{n-1}), L=[a_1,a_2,...,a_n,...].$ Is there an $a_0$ such that $L$ is dense in$[-1,1]?$
37642
Why is $\pi$ irrational if it is represented as $C/D$?
138446
Proving that there exists an irrational number in between any given real numbers
104234
What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$
43254
If $A\subseteq\mathbb N$ and $\sum\limits_{a\in A}\frac1a$ converges then $A$ has natural density $0$
87596
Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
111094
Help with proving that $\pi$ is irrational
104548
Geometric proof for irrationality of $\pi$
152172
Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
29965
Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$