How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$.   NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is not a duplicate of the mentioned question as it does not carry a correct answer and the partially correct answer in the said question is given for integer multiple case, NOT for $n \in \mathbb{N}$
Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$.    Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof.  I spend part of the day studying this question   and its answers. Only one answer is clear and give clues to solve the problem. This answer is the first one. However, this answer does not answer the question nor directly, nor the proof follows from this answer.   This answer has some mistakes, he use that $[(k_1-k_2)\alpha]=[k_1\alpha]-[k_2\alpha]$ which is not true. Consider $k_1=3, k_2=1, \alpha=\sqrt{2}$ we have $[(k_1-k_2)\alpha]=2\not= 3=[k_1\alpha]-[k_2\alpha] $. We only can assure that $[k_2\alpha]-[k_1\alpha]-1\leq [(k_2-k_1)\alpha]\leq[k_2\alpha]-[k_1\alpha]$.   Who answered said something interesting about additive subgroups of $\mathbb{R}$, but unfortunately the set $X=\{nx-[nx] : n\in \mathbb{N} \}$ is not a subgroup. Considering the additive subgroup $G=\langle X \rangle$, if we prove the part (a) of the link, we get that indeed $G$ is dense on $\mathbb{R}$ but we can not conclude that $X$ is dense on $[0,1)$.  I think this problem has not been solved.  Thanks!
Are there functions or packages in Mathematica to sort subsets in lex order, colex, revlex, etc.? Where to ask Question that can be flagged too broad or opinion based I came across the following question:    This can be easily flagged for too broad or opinion based, but I wonder on which stack-exchange site can one ask answers for such a question? A function for which one-sided limits are zero and infinity Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. Word/phrase to mean something that just happens once What is the word/phrase to mean something that just happens once? For example,     I have often been told/expected to do this and not to do that. I become aimless   and gradually lose the idea of who I am. I ask myself "What is the point?" If   life is just once thing, why can't I live my life in my own way,   the way I think is right and suitable to me? If I cannot, I would   rather live a shorter life. What causes these inconsistent dark bands in some of my photos from an indoor event? Yesterday I took photos of an event at school.  The lighting was poor so I increased the iso to 3200, shutter speed was between 80 and 250 and the aperture between 2.8 and 4.0.  Canon EOS500  about 10% of the photos got this weird horizontal black area. The position of the black area is not the same,  Sometimes it is in the middle, sometimes it is in the upper or lower part of the frame  Is this an artifact from the lighting frequency? or can it be a shutter problem? Blender worked fine before I installed a GPU to my PC. Is it the GPU? How can I fix this? I already reinstalled to the latest version. When I try to move a cube, the orange outline only moves but not the cube and I cant click it. For quite some time now I have been asking myself how the world of Game of Thrones is shaped and if the people living in that world know the shape.   In questions here on SciFi.Stackexchange concerning the unnatural seasons or the definition of a year in Westeros, it is often  that the Westerosi can measure the duration of a revolution of their planet around the sun. But that only works if they know that they actually live on a planet.   In  answer it was said that people have been travelling in ships around the world. However, I cannot remember any passage in the books supporting this. Actually I only remember evidence for the opposite:    During the  on the Iron Islands, there is a man called  who promises the Ironborn treasures from lands far to the west, if they elect him as king. However, the POV character of that chapter (I think it was Victarion Greyjoy) does not believe him and sees the madness in Gylbert's eyes - as if speaking of travelling west is crazy.  Later when he leads the attack against the Shield Islands, the Iron Fleet travels out of sight of the coast line. This manoeuvre is considered very dangerous and risky. If this is already dangerous, traveling even further west appears close to suicide.   In addition, there was a  who sailed to the far west and never returned.   establishes that the world is not a hollow world, but is not conclusive as to whether it is a planet.  Is there any evidence in the books that the World of Ice and Fire is a planet? Was it established that the Westerosi know the nature of their world? Accelerating electrons via microwaves In Synchrotrons I think they use microwaves to accelerate the electrons bundles that fly through-how does putting a microwave through a cavity accelerate an electron? I know that the Electric and Magnetic fields could and do have an effect on the electron, but if the radiation is moving parallel to the electron, wouldn't the force be perpendicular? Also, how exactly does the microwave radiation look in the vacuum chamber? Are the two ends constant nodes? Here is my problematic code:  \begin{tikzpicture} \draw  (0,0)  to [bend left=20] (3,3) node[midway] {midway label}; \end{tikzpicture}   I wanted to put a label midway of the line, but instead the label is at (0,0).  If I remove [bend left=20], the line is not bent, and the label still misplaced. 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RTL8723BE PCIe Wireless Network Adapter     nm-tool:  NetworkManager Tool  State: disconnected  - Device: eth0    Type:              Wired   Driver:            r8169   State:             unavailable   Default:           no   HW Address:        28:D2:44:81:36:0B    Capabilities:     Carrier Detect:  yes    Wired Properties     Carrier:         off   - Device: wlan0   Type:              802.11 WiFi   Driver:            rtl8723be   State:             unavailable   Default:           no   HW Address:        14:2D:27:21:8D:47    Capabilities:    Wireless Properties     WEP Encryption:  yes     WPA Encryption:  yes     WPA2 Encryption: yes    Wireless Access Points      nmcli nm:  RUNNING         STATE           WIFI-HARDWARE   WIFI       WWAN-HARDWARE   WWAN       running         disconnected    disabled        disabled   enabled         disabled       sudo rfkill list:  0: ideapad_wlan: Wireless LAN     Soft blocked: no     Hard blocked: yes 1: ideapad_bluetooth: Bluetooth     Soft blocked: no     Hard blocked: yes 2: hci0: Bluetooth     Soft blocked: no     Hard blocked: no 3: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no If a batsman hits the ball inside the ground and begins running between the wickets and now the ball is thrown back to the Fielder/Bowler/WK(FBWK)    If the FBWK catches the ball in one hand and hits the stumps with other hand before the batsman gets reached, is this a run out? If the FBWK catches the ball in hands and hits the stumps with leg or any other part of his body before the batsman gets reached, is this a run out?   This scenario can occur, when the ball was thrown far away from stumps. Proof of Funds for Canadian Visa I am a Romanian citizen.  Our co-national friends who are established in Canada will provide me and another person free accommodation and food. We are both EU citizens and we will stay in Canada 1 month.  These days we are gathering documents for our online visa application and we need to prove that we have enough money to spend there.   How much money is generally required in order to be sure we are eligible? We don't plan to spend more than 1000 EUR, but I intend to bring at least 2000 CAD. Is it enough? Deriving the inverse of a 2x2 matrix I am looking for a derivation for the inverse of a 2x2 matrix. I am also wondering why the determinant is involved in the expression. I am familiar with high school maths and linear algebra. 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