The question is stated in the title, but I don't see how to prove it. Each element of $span\{x,Y\}$ is of the form $cx + y$ where c $\in \mathbb{R}$ and $y \in Y$. If I could show that for any convergent sequence in span{x,Y} $\{c_nx + y_n\}$, $c_n$ converges to some $c \in \mathbb{R}$ and $y_n$ converges  to some $y \in Y$, I would be done. But I'm prevented from doing that because $c_n$ and $y_n$ are interdepedent. Any ideas?
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!
Prove that if $a&gt;0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $.   Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By factoring $(a^{10}-1)(a-1) \geq 0 $.  I am not sure how to proceed any further. Maximum value of $\sin A+\sin B+\sin C$? How do I set my DNS when resolv.conf is being overwritten? Most of the info I see online says to edit /etc/resolv.conf, but any changes I make there just get overridden.  $ cat /etc/resolv.conf  # Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8) #     DO NOT EDIT THIS FILE BY HAND --  #     YOUR CHANGES WILL BE OVERWRITTEN nameserver 127.0.1.1   It seems that 127.0.1.1 is a local instance of dnsmasq. The dnsmasq docs say to edit /etc/resolv.conf. I tried putting custom nameservers in /etc/resolv.conf.d/base, but the changes didn't show up in /etc/resolv.conf after running sudo resolvconf -u.  FYI, I don't want to change DNS on a per-connection basis, I want to set default DNS settings to use for all connections when not otherwise specified.  UPDATE:  I answered this question myself:   I think it's the best solution since:   It works. It requires the least amount of changes and It still works in conjunction with dnsmasq's DNS cache, rather than bypassing it. Calculate battery life I would like to know how to calculate battery life when using an LED strip. The LED strip is powered by 8 AA batteries in series. LED Current Draw = 260 mA. LED power consumption: 3.1 W. LED Operating Voltage range: 9-14.8 VDC. Please see attached images for battery datasheet.  This is my calculation, but I'm not sure if it's correct:     Battery Life = (Battery Capacity)/(Load Current) = 2000 mAh /260 mA = 7.7 Hours   This is for one battery, and the lights are powered by 8 batteries, so is the total battery life = (7.7)x(8) = 61.6 hours ? Java equivalent of unsigned long long? In C++, I enjoyed having access to a 64 bit unsigned integer, via unsigned long long int, or via uint64_t. Now, in Java longs are 64 bits, I know. However, they are signed.  Is there an unsigned long (long) available as a Java primitive? How do I use it? Similar to pissing contest - but not in an aggressive manner. Just each guy attempting to out talk one another to show they are the leader / in charge.   Sample sentence:     Well, that meeting was more of a X than being productive with Greg trying to show he was in control. How to compute the  formula $\sum  \limits_{r=1}^d r \cdot 2^r$? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you Aligning table within latex editor I just signed up, a day or two ago, and I've received a notification that there's something in my inbox.  Unfortunately, my email client isn't able to show me a link to my Inbox, or the message, because of the format of the email.  I normally use the email address that I gave when I signed up, just with MailWasher, which is intended to safely preview emails, as plain text, to decide whether or not to open them in a normal HTML email client.  I would be nice if that email contained a link to my inbox, which could be shown in MailWasher.  In order to collect that email in a normal email client, I'd also have to collect or delete all the other emails sent to that address.  I'm not entirely clear what's been done in that email to stop MailWasher seeing the link.  I can see a link to leave feedback, and one to unsubscribe, but not one to check my inbox.  Sure, I can live without one, but those emails would be a lot more useful to me if I could just click a link to go to my Inbox. I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing. What tool do I need to install to make my terminal show the autocompletion of the command that I'm writing? Why does jQuery or a DOM method such as getElementById not find the element? Does light reflect if incident at exactly the critical angle? A lot of textbooks and exam boards claim that light incident at exactly the critical angle is transmitted along the media boundary (i.e. at right-angles to the normal), but this seems to violate the principle of reversibility in classical physics.  How would a photon or ray travelling in the reverse direction "know" when to enter the higher refracting medium?  It can't know, so I conclude that such light is simply reflected?  Is this correct? Can we add a section in our vf page and add components to it Can we add a section in our vf page and add components to it. I mean split my vf page into two parts and then add components to it. Why does autoboxing make some calls ambiguous in Java? I noticed today that auto-boxing can sometimes cause ambiguity in method overload resolution. The simplest example appears to be this:  public class Test {     static void f(Object a, boolean b) {}     static void f(Object a, Object b) {}      static void m(int a, boolean b) { f(a,b); } }   When compiled, it causes the following error:  Test.java:5: reference to f is ambiguous, both method     f(java.lang.Object,boolean) in Test and method     f(java.lang.Object,java.lang.Object) in Test match  static void m(int a, boolean b) { f(a, b); }                                   ^   The fix to this error is trivial: just use explicit auto-boxing:  static void m(int a, boolean b) { f((Object)a, b); }   Which correctly calls the first overload as expected.  So why did the overload resolution fail? Why didn't the compiler auto-box the first argument, and accept the second argument normally? Why did I have to request auto-boxing explicitly? Creating Inky-looking characters in Blender I'm involved in a project to do with ink animation and experimentation. I found this incredible piece of animation:    I originally thought this was some sort of After Effects compositing (is it even possible in AE?!), but found out that this particular animation was created in 3DS Max with Krakatoa and FumeFX. I'm just wonder if a similar effect is doable with Blender's smoke simulator.  There are various YouTube videos that have pretty decent results of ink renders from Blender, but I can't seem to find anything of a similar sort of movement animation. Just blobs floating around! The closest thing I found was this animation of a         but again I can't seem to replicate this. The smoke just doesn't seem to follow my character's animation. Any help with this would be greatly appreciated. Thanks. How to influence the position of float environments like figure and table in LaTeX? How to influence the position of float environments like figure and table in LaTeX?   This is a general question and should collect useful answers for all users. I hope we can use this as a reference C++ display stack trace on exception I get the following error when i do sudo apt-get update  W: GPG error: http://ppa.launchpad.net precise Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 6AF0E940624A220   I tried following ways after reading various questions here and also by googling.  I tried  sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 6AF0E1940624A220   It did not help. I also tried  sudo gpg --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys   thinking port 11371(default) may be blocked due to firewall.  I also tried to reset everything as follows  sudo rm /var/lib/apt/lists/* -vf sudo apt-get update &amp;&amp; sudo apt-get upgrade   still i get the same error or get update. Any other possible solution I can try? Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H (\Omega, \mathcal F,P)$ of all $H$-valued random variables. My question is: How Can we define $|A|$ and the adjoint $A^*$?