Given a sequence of real numbers $a_n$ ($n \in \Bbb{N}$) that converges to $a \in \Bbb{R}$, show that the sequence $b_n$ ($n \in \Bbb{N}$) also converges to  $a$.  $$b_n=\frac{1}{n}(a_1+a_2+a_3+...+a_n)$$  I would be grateful for any tips, I have no idea where to look for sequences converging to sequences(?) - or is only $a_n$ a sequence and $a$ is not? I thought about the opposite, but couldn't help myself that way either ("if $b_n$ converges to $a$, does $a_n$ converge to $a$?"). I'd be grateful for any solutions to similar problems, that won't give away the solution - which I have in my book. I'd still like to understand it without looking that up though.
I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks!
What kind of beings are shown by the Marauder's Map? The map definitely shows people (humans) inside the castle But I remember reading Harry watching Peeves bouncing around in the trophy room and Peeves is not human And I don't remember if I had exactly read it or not, but  says that it shows the ghosts as well So what kind or creatures or beings come under the map's radar? Any living creature? Or humanoids? Or creatures with a significant size or intelligence? Would it show the house elves in the kitchen, centaurs in the forests and the merpeople in the lake as well? What about the giant squid and the thestrals or unicorns? Would it have shown the troll from the Philosopher's stone if they had the map then? What about Quirell/Voldemort? Would he have shown up on the map? And the basilisk from the chamber of secrets (the book)? May be the chamber itself was unplottable but if the map could show the basilisk he would have been visible when he was slithering around in the pipes. Why do Unix man pages use double backticks in place of double quotes? I've noticed that man pages and other documents formatted by Unix utilities often use double backticks `` followed by double single quotes '' to wrap quoted phrases instead of the double quote character ". Single quotes are similarly replaced. Why is this?  Here are a couple examples, from the man page for grep:   To find all occurrences of the pattern `.Pp' at the beginning of a line:         $ grep '^\.Pp' myfile   The apostrophes ensure the entire expression is evaluated by grep instead  of by the user's shell.  The caret `^' matches the null string at the  beginning of a line, and the `\' escapes the `.', which would otherwise match  any character.   The grep utility is compliant with the IEEE Std 1003.1-2008 (``POSIX.1'')  specification. What procedure should be followed post spilling fluids onto a MacBook keyboard? I know the first thing to do is dry with a towel the majority of the fluid. Next is to turn off the computer. Then flip over the laptop itself and whilst upside down cool dry the keyboard with a hairdryer for a few minutes. Also leave the computer upside down for a day or so. Is there anything I’m missing? I'm a trainee teacher and I need to pass a skills test in maths, but I'm having trouble in this particular area of algebra in finding the nth term of any sequence. I've been given a series of questions to answer one of them is:  Find the nth term of the sequence 3, 8, 15, 24.  I not only need to know the answer but more importantly the technique. This is GCSE level. Why are there only derivatives to the first order in the Lagrangian? Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?  Is there a good reason for this or is it simply "because it works". What is the standard error of the sample standard deviation? I read from  that the standard error of the sample variance is  $$SE_{s^2} = \sqrt{\frac{2 \sigma^4}{N-1}}$$  What is the standard error of the sample standard deviation?  I'd be tempted to guess and say that $SE_{s} = \sqrt{SE_{s^2}}$ but I am not sure. Why Does Software Updater Say No updates Available, but apt-get upgrade Shows updates available? The limit of $((1+x)^{1/x} - e+ ex/2)/x^2$ as $x\to 0$ Does name of sitemap file that Wordpress generates matter? What was Tauriel doing during the events of The Lord of the Rings movies? [Yes, The Hobbit movies are nonsense. But how could I call myself a fantasy fan if I didn't demand my nonsense be as logical as possible?]  Tauriel is presented as a highly-skilled fighter and a combatant for the side of good. Despite this, she apparently did nothing during Sauron's war on the free peoples.  Was she somehow     still mourning Killi, after all these decades?   Was she dead? Or is there another reason she didn't participate? I am trying to use a file repository for updating linux/ubuntu on a non-connected machine. The architecture for this machine (uname -a):  x86_64 GNU/Linux   The results of dpkg --printarchitecture are:  amd64   It is a Trusty Tahr amd64 machine, and it specifically says this, for example, in the aptitude sources.list file. Also, the "About this Computer" in ubuntu specifically says the machine is 64-bit. So, I installed the amd64 packages on the machine. However, when I try to do apt-get update, I get errors such as:  W: Failed to fetch file:/var/spool/apt-mirror-trusty/mirror/archive.ubuntu.com/ubuntu/dists/trusty/main/binary-i386/Packages  File not found   So, in other words, even though it is a 64-bit machine, apt is inexplicably trying to read i386 packages. Why is it doing this and how can I make it use the 64-bit packages?  Note that there is nothing in sources.list that specifies i386.  The only uncommented line in the sources.list file is:  deb file:///var/spool/apt-mirror-trusty/mirror/archive.ubuntu.com/ubuntu trusty main restricted universe multiverse Postgis, get the points that are x meters near another point, in meters Reference for the subgroup structure of $\rm{PSL}_2(q)$ This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't speak german and Dickson's treatment often requires deciphering, I was wondering if there is a "modern" account of this somewhere. Am I going to have trouble flying without my full name on the ticket? In my passport, my name is Firstname middlename1 middlename2 lastname, and in the ticket is LASTNAME/FIRSTNAME MR.  Will be a problem? I'll travel from Brazil to Bali, Indonesia and according to the agency I bought the ticket from, there's no problem and it's common.  The airline is Qatar Airlines and I have looked , which says that the passenger name can't be changed after the booking has finished.  But I don't know if it is a problem. Cycle Normal Pass in Camera Space How to determine if the specific time is between given time range in javascript What happened to Control.InvokeRequired in WPF? I am currently learning electromagnetism and I am getting really annoyed using a physical hand to accomplish the right hand rule. I get it, i've seen diagrams, and i've solved problems with it but still somehow I mess up because I bend a finger or twist my wrist a strange way. I am tired of using my hand.  So my question is can I find an entirely mathematical basis for this?  Lets say for any given diagram I define the positive $x$ axis as going to the right, the positive $y$ axis as going up and the positive $z$ axis as going into the page. Then I just did unit vectors for everything to determine the missing component?  As I understand it $\vec{F} = q (\vec{V}\times\vec{B})$ and with that definition I am confident I could find out the direction of the electric field given the velocity of a particle of charge and the magnetic field but how can I re-arrange this so I can solve for other variables? For example what does $\vec{B}$ equal in terms of a cross product involving $\vec{V}$ and $\vec{F}$? Why is my text pixelated? Within my game I have some cubes with text which render fine. In addition to the cubes I want to display some static text, e.g. the current score. The score text has the camera as parent and a copy rotation constraint on the camera. While this concept works fine in principle, the score text is displayed very pixelated. I already increased the font size to 50 and made the text smaller by scaling it down, but it still doesn't show nicely. The score text is about 11 Blender units away from the camera. If I move the score text further away, it displays better, but then the text interferes (collides) with the cubes. What could I do to improve the quality of this text? Download . Open the SSCCE in Blender 2.70. Hover your mouse over the 3D viewport. Press P to start game engine. How to automaticaly enter sudo's grace period upon login