Let $X$ be a countable set. Then which of the following are true?   There exists a metric $d$ on $X$ such that $(X,d)$ is complete There exists a metric $d$ on $X$ such that $(X,d)$ is not complete There exists a metric $d$ on $X$ such that $(X,d)$ is compact There exists a metric $d$ on $X$ such that $(X,d)$ is not compact   My attempt:-  1.$(\mathbb N,d)$, where $d$ is a discrete metric space is complete   $(\mathbb N,d)$, where $d(x,y)=|x-y|$ is not a complete I am not able to judge $(\mathbb N,d)$, where $d$ is a discrete metric space is not compact.
Let, $X$ be a countable set. Which of the following are true?   There exists metric $d$ on $X$ so that $(X,d)$ is complete. There exists metric $d$ on $X$ so that $(X,d)$ is not complete. There exists metric $d$ on $X$ so that $(X,d)$ is compact. There exists metric $d$ on $X$ so that $(X,d)$ is not compact.   4 is true as we can take discrete metric. Then $\{\{x\}:x \in X\}$ is an open cover containing no finite subcover.  If there exists a compact metric then it would automatically become complete also. Hence if 3 is true, so is 1.  What about 2? Any other comment?
Pointwise limit of continuous functions not Riemann integrable Infinite series for $e$... How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this... Set limit to one author when using "et al." in biblatex How do you serialize an object in C++? I have a small hierarchy of objects that I need to serialize and transmit via a socket connection.  I need to both serialize the object, then deserialize it based on what type it is.  Is there an easy way to do this in C++ (as there is in Java)?  Are there any C++ serialization online code samples or tutorials?  EDIT: Just to be clear, I'm looking for methods on converting an object into an array of bytes, then back into an object.  I can handle the socket transmission. Where, precisely, does feedback on Community ♦ generated flags go to die? For the most part, I treat "Community" like any other user when handling system raised flags. If they are useful, I mark them as such. If they are noise, I dismiss them as such. The Community user has a static flag weight, so our reaction to them is more or less feedback for developers.  Is this feedback from moderators network wide used in any way to help improve the accuracy of system raised flags? Or, would clicking either button suffice?  I'm asking because our reasoning behind declining a flag is now more precise. Does dismissing some as noise help improve the criteria that ultimately generates the flag?  Edit  Yes, thank you Skype, I realize that  asked a similar question . I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks! How can I get a reference to a method? How to measure correlation between multi-item likert scales? I recently conducted a online survey and are now having trouble deciding for the right analysis method. For example: I wanted to see if there is a correlation between Factor A and Factor B. However in my survey I did not ask the participants to directly rate Factor A and Factor B, but rather asked multiple subquestions for these two factors which were rated from 5(Strongly Agree) to 1 (Strongly Disagree). I created a sum score for item A and a sum score for item B by summing up the values of each subquestion and calculated the mean of the sumscores to measure them. So my question ist: Which method would be best to use here? I want to test if e.g. people with a higher score in Factor A would be more likely to decide for Factor B. I actually planned using Spearman or ANOVA depending on the number of independent variables, but I am not sure if that would really be the correct way to measure the correlation. Introduction to proofs with a fair amount of hand-holding? Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff on his own.  He asked me recently if there was a 'mathematical' version of .  The point being, if there was an "introduction to proofs" book which took a more hand-holding approach. He's also expressed fondness for the .  However, metamath's proofs are obviously a bit too low level to learn anything serious.  At one point in our conversation, I mentioned the dependently typed programming language , but I felt it's not quite as helpful until he at least gets some experience with  (for which the target audience of Agda is Haskell programmers, hence the similarity in syntax).  Anyone have any good suggestions on a book? How do the comment replies (aka &quot;pinging users&quot;) work?  Who can I send an @reply to? How do I respond to a specific user when entering my comment? Will they be notified? What do people mean when they talk about &quot;pinging&quot; another user?  Related: QGIS 2.0 - can't select items in Print Composer Problem from Herstein on group theory The problem is:  If $G$ is a finite group with order not divisible by 3, and $(ab)^3=a^3b^3$ for all $a,b\in G$, then show that $G$ is abelian.  I have been trying this for a long time but not been able to make any progress. The only thing that I can think of is: $$ab\cdot ab\cdot ab=aaa\cdot bbb\implies(ba)^2=a^2b^2=aabb=(\text{TPT})abba.$$ Now, how can I prove the last equality? If I write $aabb=abb^{-1}abb$, then in order for the hypothesis to be correct, $b^{-1}abb=ba\implies ab^2=b^2a$. Where am I going wrong? What should I do? What article should be used in the following sentence?     He was English by [a/the/] blood.   I feel there should be a zero article here, but I was taught that the zero article is impossible in English. Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$ I think we use trig substitution for this. But I'm not sure where to even begin with this   $$\int \frac{25 - x^2}{x^4} \operatorname d\!x$$ Unable to catch JsonMappingException How come unpolarized light does not undergo destructive interference? Let $x \in \mathbb R$  Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ .   Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher was not able to give a valid solution without  resorting to Fourier series.   He actually wanted to prove first that the function is a solution of the ODE $$ y-y'' =0$$ without succeeding in proving it.   Anyway the result sould be $\dfrac{\pi}{2}e^{-x}$  Thanks for any help on this subject. Is there a drupal.org API for retrieving the list of available modules? Moderncv package - cventry date width I am writing my CV using moderncv package, everythig is nice but I have a problem with \cventry command. I have something like this:   (it means 2010 - now)  I would want to have this in one line, rather than in two lines. I tried looking into the moderncv style files but with no success. What can I do to achieve the effect I want? I'm trying to install a service using InstallUtil.exe but invoked through Process.Start. Here's the code:  ProcessStartInfo startInfo = new ProcessStartInfo (m_strInstallUtil, strExePath); System.Diagnostics.Process.Start (startInfo);   where m_strInstallUtil is the fully qualified path and exe to "InstallUtil.exe" and strExePath is the fully qualified path/name to my service.  Running the command line syntax from an elevated command prompt works; running from my app (using the above code) does not. I assume I'm dealing with some process elevation issue, so how would I run my process in an elevated state? Do I need to look at ShellExecute for this?  This is all on Windows Vista. I am running the process in the VS2008 debugger elevated to admin privilege.  I also tried setting startInfo.Verb = "runas"; but it didn't seem to solve the problem.