So I am having trouble proving that if  $$ K= \{0\} \cup \left\{\frac1n: n = 1,2,\ldots\right\} \cup \left\{\frac1n + \frac1m : n=m,m+1,\ldots; m=1,2,\ldots \right\} $$  then 0 and the points $\frac1m$ are the only limit points of $K$. Cooke proves this by saying that:  Since $x\geq0$ for all $x \in K$ and for any positive number $\epsilon$ there is only a finite set of number s in $K$ larger than $1+\epsilon$, it is clear that no negative number and no number larger than 1 can be a limit point of K. Hence we need only consider positive numbers $x$ satisfying $0\lt x \lt 1 $. If $x$ is such a number and $x$ is not one of the points $\frac1m$, let $p$ be such that $\frac1{1+p} \lt x \lt \frac1\epsilon$, and let $\epsilon=\frac12\min(x-\frac1{p+1},\frac1p - x)$.  The intersection of the set $K$ with the interval $(x-\epsilon,x+\epsilon)$ is contained in the set of points  $$ \left\{\frac1{p+1}+\frac1k:p+1\le k \lt \frac1\epsilon\right\} \cup \left\{\frac1m+\frac1n:m\le n \lt \frac1{p+1} - \frac1{p+2}; m=p+2, \ldots,2p+2\right\}, $$  which is a finite set. Therefore $x$ cannot be a limit point of $K$.  I understand why you take $\epsilon$ to be half of $\min$, and the using that every neighborhood around a limit point contains infinitely many points, but I don't understand why the intersection of $K$ with the interval $(x-\epsilon,x+\epsilon)$ is contained in the set mentioned above, and why it is finite. Is it a typo? $$ \left\{\frac1m+\frac1n:m\le n \lt \frac1{p+1} - \frac1{p+2}; m=p+2, \ldots,2p+2\right\}, $$  here I think n is meant to be an integer in order for the finite argument to work, but how can be  $$ n \lt \frac1{p+1} - \frac1{p+2}\:\:? $$ Here is how I fixed the typo to work: if  $$ \left\{\frac1m+\frac1n:m\le n \lt \frac1{p+1} - \frac1{p+2}; m=p+2, \ldots,2p+2\right\} $$  is changed to  $$ \left\{\frac1m+\frac1n: m\le n, \frac1n \ge \frac1{p+1} - \frac1{p+2}; m=p+2, \ldots,2p+2, n \in N\right\}, $$  then $m\le n \le (p+1)(p+2)$ which would be finite. Am  I understanding this argument correctly? I think there is a typo or error in Cooke's solution, as the original version can't seem to work.  I know there are other constructions that can be understood easily, but I would like to understand Cooke's proof of $\frac1n + \frac1m$ and see if there are are any typos or errors in the original solution. Thanks a lot.
I need to find limit points of the set $\{  \frac{1}{n} +\frac{1}{m} \mid n, m = 1,2,3,\dots \}$.  My try :   If both $m$ and $n$ tend to very large values say $\infty$ then the value of $\{ \frac{1}{n} +\frac{1}{m} \}$ tends to $0$, and if only one of $m$ or $n$ tends to very large values, then the set $\{ \frac{1}{n} \mid n=1,2,3,\dots\} $ acts as limit points.   So is it true that a set of limit points is $\{0\}$ $\cup$  $\{\frac{1}{k} \mid k=1,2,3,\dots \}$?  How should I write this proof rigorously?
"I want you succeeding" vs. "I want your succeeding" I am happy seeing you succeed.   I am happy to see you succeed.   I want you to succeed.   I want you succeeding. I want your succeeding.  Could you please explain the differences between them, and the first and second examples independently of each other? What does "can only heal so much" mean in this sentence? I found the following sentence in an American novel (this one:  you can read the whole paragraph). A woman broke her wrist and just had her cast removed. She thinks she can start playing piano again, but her wrist still hurts too much. I'm not sure I understand the meaning of the final sentence:  But a body can only heal so much, so fast.  Can someone help me, please? Can you help me with my software licensing issue? This is a  about Licensing.    Questions on licensing are off-topic on Server Fault.  If your question has been closed as a duplicate of this question, then this is because we want to help you understand why licence questions are off topic rather than just telling you "it just is". In all likelihood, this question will not address your question directly, it was not meant to.    I have a question regarding software licensing. Can the Server Fault community please help with the following:   How many licenses do I need? Is this licensing configuration valid? What CALs do I need to be properly licensed? Can I run this product in a virtual environment? Can I downgrade this product to an earlier version? Am I entitled to feature X with license Y? How can I prevent my mail from being classified as spam? Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$ Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof? Triangular Factorials I came across a statement online and have been looking for a proof :  It states that 1, 6 and 120 are the only numbers which are both triangular and factorials.  Is there any way I can prove this? This claim looks too 'big' and I've tried to prove it but  I couldn't. Can anyone help me to prove this? User’s Guide vs Users’ Guide I’ve been looking over what has been posted regarding the use of ’s.   I used to be a Technical Writer (years ago). The title of one of our training documents was Users’ Guide.  Once, a coworker said every time he saw that title he expected users to start coming.    Never made any sense to me, but I have to admit that the majority of us didn't understand the use of s’ in place of ’s.  What is the difference? Say I had a Core i7 Laptop with 16GB of RAM and a 750gb drive.  Is there a mechanism where I can partition the HD into a Linux, Windows, OS X, Shared Data, Hypervisor setup where I boot a small hypervisor. I can start/stop a linux, Windows and OS X virtual machine and interact with them all from the hypervisor?   Code, Test, etc?  is this a type 1 Hypervisor?  I suppose I could run a small linux install and VmWare but could VmWare read the virtual machines from a physical hard disk partition versus a file based setup (.vmdk files)? scp copy over ssh doesn't work - permission denied error, please? In chrome, most of the time, when i middle click on a link (text or image), it opens a new tab and the current tab does not change. This is the expected behaviour.  However, some websites have links (i'd say 5%) that also switches the current tab to the new one. For example: on this , the name of the players in the main list have the problem.  By looking at the source, i didn't find anything anormal (the link is a standard html tag). But if i disable javascript, i have not the problem. I looked at the source, and didn't find the problematic script... Note also that if i open the tab with right click > open link in new tab, there is no problem.  I'd like to know if you find the reason of the problem and how i can fix it. I am running chrome 10.0.648.204 on windows 7 x64.  EDIT: the website above does not exhibit the behavior anymore, but I sill have the problem sometimes on other sites (which I don't remember) Warn new users when they ask a question after a previous question is closed, downvoted, or deleted Another &quot;miséricorde&quot;:  Consider  (emphasis added):  I think its okay, please dont take any efforts for me. I personally do not feel the system helps ask good questions. I dont know why asking for a software in this case is bad. Basically I needed a command that would give me this data. There is no other way I could have asked the question. I could have changed the question just for the sake of changing. I feel I am better off elsewhere. Thanks for your very very prompt responses. Regards. Bye. – user1744649  We just lost another user due to inadequate guidance on what questions are allowed.  The  by the time they are banned has been proven incorrect. Given that , we need a better way to make sure that users can understand how to ask good questions and what types of questions are allowed. I suggest that we warn newcomers, in a clear but positive tone, when one of their questions has been closed, downvoted (to a score of -2 or lower), or deleted, so that they don't repeat the same mistake.  This is probably best done the next time the user asks a question, and should appear as an &quot;interstitial&quot; page prior to reaching the actual Ask Question form.  The page should not appear to users:  with at least 50 (or perhaps 75 or 100) reputation that have asked a good question since the last downvoted/closed/deleted question that have fixed a bad question and had it reopened/undeleted/upvoted.  The content of the page should be adapted based on the specific problem with the previous question.  I would imagine something similar to the following (example for a question closed as off topic on Super User):  Before you ask another question... One or more of your previous questions have been closed as off topic. The scope of on-topic questions is defined in the , as follows:  What kind of questions can I ask here? Super User is for computer enthusiasts and power users. If you have a question about …  computer hardware, computer software, or personal and home computer networking  and it is not about …  programming and software development, video games or consoles, websites or web services like Facebook, Twitter, and WordPress, electronic devices, media players, cell phones or smart phones, except insofar as they interface with your computer, issues specific to corporate IT support and networks, asking for a shopping or product recommendation,  … then you’re in the right place to ask your question!  Please make sure that the question you are about to ask is on topic as described above. If you need help determining whether your question can be asked on this site, you can ask for advice on  or . You may also want to edit your previous question to address any problems associated with it. To ensure a high standard of content quality and prevent disruption to the community, users who repeatedly post low-quality questions  Continue  The  doesn't address this issue as it is shown only to users with 10 reputation or less (rather than up to 50 and possibly higher) and does not address the specific problem with previous low-quality questions.  The point of this feature request is to provide a easily understood targeted message that actually teaches the user about what happened with the previous question he or she asked.  The EULA is generic and may not have conveyed the intended message clearly if the user still asked a low-quality or off-topic question. For reference,  currently has 143 reputation and the proposed message would probably have been shown after the first closed or deleted question. I know this can be a complex feature to implement, since it needs to be adaptive, but this should help eliminate confusion on what the rules are and minimize the incidence of good-faith users getting post-banned. Currently in my last year of high school, and I have always been told that  does not exist by my physics teachers. Today my girlfriend in the year below asked me what centrifugal force was, I told her it didn't exist, and then she told me her textbook said it did, and defined it as "The apparent force experienced towards the outside of a circle is the centrifugal force and is due to the mass of the object resisting the inward centripetal acceleration that the object is experiencing". I was pretty shocked to hear this after a few years of being told that it does not exist.  I did some reading and found out all sorts of things about pseudo forces and reference frames. I was wondering if someone could please explain to me what is going on? Is it wrong to say that centrifugal force does not exist?   This has always nagged me a bit as I often wonder that if every force has a reaction force then a centripetal force must have a reaction centrifugal force, but when I asked my teachers about this they told me that centrifugal force does not exist. How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic? Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed.  I'm having trouble in showing $F+G$ to be closed. I know that $F$ is itself closed and complete, as it is a finite dimensional subspace of a normed space, and that if $F$ were compact that $G+F$ would be closed. I also know that the closed unit ball of any finite dimensional normed space is compact.  I tried two methods. One was to take a convergent sequence $(x_n)_{n \in \mathbb N}$ in $G+F$. Then we can write $x_n = f_n + g_n$ where $f_n$ and $g_n$ are sequences in $F$ and $G$ respectively, and I attempted to find a way to force the individual components $f_n$ and $g_n$ inside the unit ball which would enable me to say that they had convergent subsequences. I couldn't see how to do this, however.   The other thought was to try and show that $F$ is compact, but I don't see a way to do this as I can't imagine it to be simply true without some other conditions on $F$.  Are one of these methods the right way to go? Or should I go another direction? I would appreciate any help I can get, although I would prefer not to be presented with a full proof so that I can do some work for myself. Thanks! Glassware with scratched/etched surfaces generate more bubbles (, ). Why these rough surfaces allow easier heterogeneous nucleation? What is the actual mechanism behind the reflection of various colored light by material of different color? When light (composed of photons with different frequency and wavelength) falls on a material, some of it gets absorbed and some get reflected depending upon the color of the material on which it strikes. And the material reflects the light of its own color only, thereby absorbing all the other colored light. So, my questions are:  How would a photon with certain wavelength (say 400 nm) decide to get reflected after striking the material of red color?   What is the actual mechanism behind the reflection of various colored light by the material of different color? Maximal inequality for a sequence of partial sums of independent random variables Let $X_n$, $n=1,2,3,...$ be a sequence of independent (not necessarily identically distributed) random variables, let $S_n=\sum_{i=1}^nX_i$. Prove the following maximal inequality: For all $t&gt;0$,$$\mathbb{P}\left(\max_{1\leq i\leq n}|S_i|\geq t \right)\leq 3\max_{1\leq i\leq n}\mathbb{P}\left(|S_i|\geq\frac{t}{3} \right)$$ In the sentence below, are the usages of apostrophes correct?     In research about parent stereotypes, the correlation between parents’ beliefs about their early elementary age children’s’ mathematics abilities and the children’s actual abilities increased as children aged. "Heard me [infinitive]" vs. "heard me [present participle]" launch a VM on Ubuntu On Ubuntu 14.04, I installed VirtualBox on which I have created 2 Ubuntu virtual machines.  Does Ubuntu provide a way to launch  the virtual machine I want at a particual moment ? Is there any solution for this problem by developing a .sh script for example or using any other programming language ?