Define a subset $X$ of $\mathbb{R}^n$ to have property C if every sequence with exactly one accumulation point in $X$ converges in $X$. (Recall that $x$ is an accumulation point of a sequence $(x_n)$ if every neighborhood of $x$ contains infinitely many $x_n$.)  Show that any subset $X$ of $\mathbb{R}^n$ satisfying property C is compact.  Since $\mathbb{R}^n$ is Euclidean, compactness $\iff$ closed and bounded. To show $X$ is closed, we can take any limit point $x$ of $X$. By definition of a limit point, we can find a sequence in $X$ converging to $x$. Then $x$ is the unique accumulation point of the sequence and hence by hypothesis $x\in X$  How can I show that $X$ is bounded?  Alternatively, I could show that $X$ is sequentially compact, but I don't know how to proceed.
I found the following question about compactness:     A set $S\subseteq \mathbb{R}^n$ has the following property:   if a sequence of $S$ has exactly one accumulation point in $S$, then it converges in $S$. Prove that $S$ is compact.    We know that in $\mathbb{R}^n$ compact sets correspond to closed and bounded sets.   (1) I am trying to prove that $S$ is closed. The condition in the question says if a sequence of $S$ has an accumulation point in $S$ then it converges in $S$. I think instead the condition should be if a sequence of $S$ has an accumulation point( in $\mathbb{R}^n$, not necessarily in $S$ )then it converges in $S$.  Am I correct?  (2) How to prove that $S$ is bounded?  P.S. I have made an edit (there is exactly one accumulation point).  Source of the question:  of this problem set.
How often do ewoks eat sentients? In Return of the Jedi, the ewoks plan on eating Luke, Han and Chewie.     THREEPIO: I'm rather embarrassed, General Solo, but it appears you are to be the main course at a banquet in my honor.   At this point, the tribe had already met and befriended Leia, and had encountered the Imperials in the past, making it fairly likely that they knew humans were .  This raises the question of how often they ate sentients. Was it a one-off thing, to honor an extremely rare visit from a god, or was this fairly common? What is the EV scale? I have seen an "EV" (exposure value) scale that's sometimes used to express a camera's exposure settings, or scene brightness. How exactly does this scale work? Invalid token 'while' in class, struct, or interface member declaration in very simple code I am not sure what the problem is but I keep receiving this error when I try to use a while statement in my code.     Invalid token 'while' in class,   struct, or interface member   declaration   I want to use a while loop to have something continuously update while a statement is true.  The rest of my code is rather long but whenever I type in the syntax:  while(a&lt;b) { //do whatever i want it to do here }   It gives me that compiler error right off the bat. Not quite sure what the problem is. I am doing this in a C# windows application under the Form1.cs file with all the other event handlers (for buttons and such).  Thanks!    I was unaware that loops had to be placed within a method (fairly new to c#), but I tried it and no errors were returned. Thanks for your help everybody!  Previously, I just had the loop within the main class of the program. Elected moderators can nominate themselves , and I can nominate myself again, despite having won the first round of election.    I understand this isn't going to be easy to fix, given that SE only knows about community mods, employees and developer roles, and pro-tempore mods need to be able to nominate themselves. I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? How do I see my name in the panel in Gnome Shell? I have Ubuntu GNOME 14.04 LTS and from installation, my name has not appeared in the panel. I have enabled it via dconf in org.gnome.desktop.privacy, but it has yet to appear even after a re-login and restart.  Can I receive some assistance into the cause of this and a solution?  This is where I would like it to appear: How can I install Ubuntu encrypted with LUKS with dual-boot? The Ubuntu installation disk has an option to install Ubuntu encrypted using LUKS. However, there is no option to perform an encrypted installation along-side existing partitions for a dual-boot scenario.  How can I install Ubuntu encrypted alongside another partition from the live disk? I'm currently working on a scene containing 1000+ objects which all need to be set to a different draw type to save on computationally expenses in the viewport. But doing this for every object separately is really tedious. Therefor I'm wondering if there's a way to change the maximum draw type of multiple objects simultaneously. Preferably only the selected objects. "going to come down" vs "going to have to come down" What is the difference between the following sentences?        This fence is going to come down.   This fence is going to have to come down.      I have doubt that when you are saying fence is going to have to come down, that is enough to say it right? but why I hear in movies saying fence is going to have to come down, is there a difference between them? How to UPSERT (MERGE, INSERT ... ON DUPLICATE UPDATE) in PostgreSQL? Shortest path from points in one layer to points in another layer via roads in QGIS The result set has no current row while using JDBC I am using the following code in selenium, but I am not able to retrieve data from database. I am getting the errors "The result set has no current row"  My code is   public class Auto {      public static void main(String[] args) throws InterruptedException, SQLException, ClassNotFoundException, InstantiationException, IllegalAccessException     {           Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver");         System.out.println("Driver loaded");           Connection con=DriverManager.getConnection("jdbc:sqlserver://192.168.*****\\SQLEXPRESS/*****_DW/EPS:****;user=******;password=******; ");          System.out.println("Driver loadedd");       java.sql.Statement stmt= con.createStatement();     ResultSet rs=stmt.executeQuery("select * from [*****_DW].[dbo].dim_grantee ");      System.out.println("Query executed");      while (rs.next());     {         String Name=rs.getString("NAME");          System.out.println("Name is");     }   } Equicontinuity on a compact metric space turns pointwise to uniform convergence I know that   If $\{f_n\}$ is an equicontinuous sequence,  defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly.  I'm having trouble proving this.  I see the same problem  but am having trouble following the proof, particularly with part (3).  Can someone guide me through a proof of this result? Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set.  $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ where $L_{v,x_0} := \{x_0 + tv : t \in \mathbb{R}\}$.  i.e. f is convex iff when restricted to any line, it be convex on that certain line.  So essentially to handle this proof I need to show that convexity for $f$ holds when its convex over any arbitrary line, and on the converse it does not hold when $f$ is not convex over that arbitrary line. If $f$ is convex then,  $$\forall x,y \in C, a \in [0,1]$$ it holds that $$f(ax+(1-a)y) \leq af(x) + (1-a)f(y) \space \space\space\space\space\space\space\space\space\space\space\space (1)$$  EDIT: $$**Part II**$$  Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.   I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$  $\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.  where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$  Any help is very much appreciated How should sockpuppets be handled on Stack Exchange? A high-reputation, highly-active user has already been dinged once for a sockpuppet, and I just dinged him again. He probably has another sockpuppet as well.   What are the guidelines for handling users who game the system in this fashion? Specifically,   How can I be sure I'm looking at a sockpuppet? When should sockpuppets be considered a problem? How should moderators handle problematic sockpuppets once they've been identified? I was suspended for sockpuppetting and feel this is unfair relation between $R^2$ of simple regression and multiple regression A very basic question concerning the $R^2$ of OLS regressions   run OLS regression y ~ x1, we have an $R^2$, say 0.3 run OLS regression y ~ x2, we have another $R^2$, say 0.4 now we run a regression y ~ x1 + x2, what value can this regression's R squared be?   I think it's clear the $R^2$ for the multiple regression should be no less than 0.4, but is it possible for it to be more than 0.7? The HDD I have my Ubuntu installed is about to fail. I would rather not lose 3 years worth of data, customisation and apps. I am looking for a way to move the complete system (SWAP included, because I'm not sure if I can relink the system to a new SWAP partition) to another HDD. But not the complete HDD&lt; only the partition containing Ubuntu, to a partition on a different HDD. Basically I'd like to do what I've been able to do with Norton Ghost for my Windows install. I thought about using Clonezilla but I think I would have issues with GRUB (Especially trying to boot from a different UUID than what is in the conf file). do you know of any way this could be done?  PS, my home directory is encrypted but that's not really an issue, because I can work around that.  EDIT: changed the explanation to make it clearer What's better here - a single column or multicolumn primary key? Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste? Is it normal for DC wall adapter power supplies to provide voltages above the nominal ones? When I measure a 9V DC wall adapter power supply with a multimeter it shows 11.8V. When connected to my board this voltage lowers to 10V. Is this normal? Should I be afraid of toasting my  supplied board when using this adapter?  Edit: This regulator accepts up to 12V inputs. So that will be no problem. A second "9V" adapter yields 15V to the multimeter. Will this one burn my circuit? Help with proving that $\pi$ is irrational