I want to show that there is no proper subgroup of $\mathbb Q$ of finite index.  I found many solutions using quotient group idea.   But I didn't learn about that. So I want to solve it without using that.  For example I solve [$\mathbb{Q}:\mathbb{Z}$] is infinite like this.  Suppose $[\mathbb{Q}:\mathbb{Z}$] is finite.  Then $\exists a_1, \ldots , a_n $ s.t. $ \mathbb{Q} = \bigcup_{i=1}^n (a_i+ \mathbb{Z}) $  WLOG, 0$\le a_1, \ldots , a_n \lt $1  Since $\mathbb{Q} $ is dense, $\exists 0\le r \lt 1$ s.t $ r\neq a_i $ for each i.  Then $r\in \mathbb{Q} $ but $r\notin \bigcup_{i=1}^n (a_i+ \mathbb{Z})$, contradiction.  Can I solve my question likewise?
Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$.  I just saw this on our exam earlier and was stumped on how to show this. Any ideas?
Reading over the Monk I saw that for Martial Arts:     When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.   This implies that non-monk weapons do not count towards being able to make this single unarmed strike. However, when looking over Flurry of Blows:     Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.   The requirement doesn't seem to be there anymore, and I didn't see where this discrepancy was covered in . This  says that you can use flurry of blows as a bonus action on any form of attack.   So does the limitation really only exist on a single unarmed strike, but you can use whatever attack action you want to trigger Flurry of Blows? Conjugate class in the dihedral group List all the conjugate classes in the dihedral group of order $2n$ and verify the class equation.  The dihedral group is generated by two elements $r$ and $s$.   The order of $r$ is two since $r^2=e$ and $s$ is $n$ since $s^n = e$. And I know all elements can be produced as either $s^k$ or $rs^k$. How can I list all the conjugacy class? How do I loop through or enumerate a JavaScript object? I have a JavaScript object like the following:  var p = {     "p1": "value1",     "p2": "value2",     "p3": "value3" };   Now I want to loop through all p elements (p1, p2, p3...) And get their keys and values. How can I do that?  I can modify the JavaScript object if necessary. My ultimate goal is to loop through some key value pairs and if possible I want to avoid using eval. Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$ Non-hidden directory doesn't show in Explorer, but the program that made it can access it Mail vs. mail what is the difference If router is off for a few days, do you have an ip address as far as ISP is concerned? I have an ISP which assigns IP addresses dynamically. If I go out and turn my router off I am always assigned a new one once I turn it back on. Does that mean that...  1) my previous IP goes elsewhere when my router is turned off?  2) i dont actually have an IP address if my router is off for a day or more? I really read many tutorials, but crontab doesn't start notify send.  What I've tried:   etc/crontab  added at the end:  15 * * * * notify-send 'crontab' 'test'  with crontab interface:  crontab -e   added at the end:  15 * * * * notify-send 'crontab' 'test' Casting vs using the 'as' keyword in the CLR When programming interfaces, I've found I'm doing a lot of casting or object type conversion.  Is there a difference between these two methods of conversion?  If so, is there a cost difference or how does this affect my program?  public interface IMyInterface {     void AMethod(); }  public class MyClass : IMyInterface {     public void AMethod()     {        //Do work     }      // Other helper methods.... }  public class Implementation {     IMyInterface _MyObj;     MyClass _myCls1;     MyClass _myCls2;      public Implementation()     {         _MyObj = new MyClass();          // What is the difference here:         _myCls1 = (MyClass)_MyObj;         _myCls2 = (_MyObj as MyClass);     } }   Also, what is "in general" the preferred method? utf8 and ISO 8859-1 encoding references trouble I have to use one particular template ( am obliged to use it for my thesis) it uses \bibliographystyle{plain} Ihave both english and frensh text on it (for abstract and résumé) there is no problem for characters like ( é, ç...) but some of references does not appear they are replaced by [?] knowing that foe encoding \RequirePackage[latin1]{inputenc} how to fix this problem please!   \documentclass{article}   \RequirePackage[latin1]{inputenc}   \RequirePackage{fancyhdr}     blabla \cite{nayak2015comprehensive, byun2002applications, shah2004sequential, fletcher2009support, cornuejols2011apprentissage, byun2002applications}   @article{nayak2015comprehensive,    title={A comprehensive survey on support vector machine in data mining     tasks: applications \&amp; challenges},    author={Nayak, Janmenjoy and Naik, Bighnaraj and Behera, HS},    journal={International Journal of Database Theory and Application},    volume={8},    number={1},    pages={169--186},    year={2015}    }      @inproceedings{shah2004sequential,     title={Sequential k-nearest neighbor pattern recognition for usable      speech classification},     author={Shah, Jashmin K and Smolenski, Brett Y and Yantorno, Robert E      and Iyer, Ananth N},     booktitle={Signal Processing Conference, 2004 12th European},     pages={741--744},     year={2004},     organization={IEEE}     }      @misc{fletcher2009support,      title  = {Support vector machines explained},     author = {Fletcher, Tristan},     journal= {University College London, London},     year   = {2009},     howpublished = "\url{http://sutikno.blog.undip.ac.id/files/2011/11/SVM-     Explained.pdf}",     note   = {Accessed: 2017-04-30}     }      @inbook{cornuejols2011apprentissage,      title={Apprentissage artificiel: concepts et algorithmes},      author={Cornu{\'e}jols, Antoine and Miclet, Laurent},      year={2011},      pages={279--309},      publisher={Editions Eyrolles}       }      @incollection{byun2002applications,      title={Applications of support vector machines for pattern recognition:       A survey},      author={Byun, Hyeran and Lee, Seong-Whan},      booktitle={Pattern recognition with support vector machines},       pages={213--236},      year={2002},      publisher={Springer}      }    \end{document}  knowing that my references work well on other template but this one no how to fix this problm please! here is the error I got   Edited here is the link to the template  in fact I have created a littele example ( since I could not publish my thesis ...) the error has disappeared but my references still [?] Product and Box Topologies I am having a hard time understanding why the box topology is finer the the product topology. Of course I know that with finite product, the two are the same. With product topology, the basis elements are defined so that the $U_{\alpha}=X_\alpha$ except for finitely many $\alpha$. So how can this topology be contained in the box topology which has only the product of the  $U_{\alpha}$ as basis. I think it should have been the other way round. Am I missing something here? I need help with this because I have been confused over it for a long time. I don't like Hell, can I go somewhere else? I just opened a nether portal for the first time, and I must say, it's not a very good location, a sort of narrow trench/cavern thingy without any obvious clues as to which direction is "out".  I'm not willing to simply move my overworld portal (due to dilation it is unlikely to have much of an effect, anyway), and was wondering if there was a way to force minecraft to generate a new personal Hell for me to explore (hopefully one that doesn't involve me stuck in some wall!). Is JavaScript's "new" keyword considered harmful? In another , a user pointed out that the new keyword was dangerous to use and proposed a solution to object creation that did not use new. I didn't believe that was true, mostly because I've used Prototype, Scriptaculous and other excellent JavaScript libraries, and everyone of them used the new keyword.  In spite of that, yesterday I was watching Douglas Crockford's talk at YUI theater and he said the exactly same thing, that he didn't use the new keyword anymore in his code ().  Is it 'bad' to use the new keyword? What are the advantages and disadvantages of using it? Is http://stackexchange.com/ down ? Direction of $d\mathbf{l}$ A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help! Very basic question here, I am afraid.  I have hundreds of shattered pieces of a mesh and I am trying to animate their visibility over the scene.  How can I animate the visibility and render visibility parameter for one piece, then copy/transfer to all the others? How to prevent Google from finding my admin index page? Formula based on picklist I am trying to create a formula that will put a 1 in a hidden field if the value of a picklist is equal to a specific value.  I keep getting stuck here and I can't figure out what I am missing?  IF(ISPICKVAL( Examiner_Type__c ="Pending Examiner Status","1","",)) install other packages when creating new package in R What does "rigorous proof" mean? I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance.