Let $f:[0,1]\rightarrow [0,1]$ be a continuous function and $x_1\in [0,1]$. If $x_{n+1}= \frac{\sum_{i=1}^n f(x_i)}{n}$ then prove or disprove that $\{x_n\}$ is convergent. I have proved  If $f$ is constant then the limit exists.  $\{(n-1)x_n\}$ is monotonic.  By compactness $\{x_n\}$ has limit points   But I am unable to show whether the sequence is monotonic or not and it has one limit point or more than one limit points. I come to know that a similar question was discussed in the following link  But some steps in that solution are not clear to me
I have tried a little bit which as follows- Since $f(x_n)\in[0,1]$, $\{f(x_n)\}$ has a convergent subsequence say $y_n=f(x_{r_n})\ \forall n\in\Bbb{N}$ Let, $\lim y_n=l\implies \lim \frac{y_1+y_2+\cdots+y_n}{n}=l\implies \lim \frac{f(x_{r_1})+f(x_{r_2})+\cdots+f(x_{r_n})}{n}=l$ But I am getting no idea to proceed and prove the convergence of $\{x_n\}$. I have also tried to prove the sequence to be cauchy which goes- $x_{m+1}-x_{n+1}={\sum_{i=1}^m f(x_i)\over m}-{\sum_{i=1}^n f(x_i)\over n}\le {\sum_{i=1}^m f(x_i)\over n}-{\sum_{i=1}^n f(x_i)\over n}$ (since $m\ge n)$ $\implies |x_{m+1}-x_{n+1}|\le \frac{|f(x_m)|+|f(x_{m-1})+\cdots+|f(x_{n+1})|}{n}\le\frac{m-n}{n}$ (since $f([0,1])\subseteq [0,1])$ Now, what will I get if I tend $m,n\to\infty$? But I need to do something more in the 2nd case since I have nowhere used the continuity of $f$.  Can anybody give an idea to prove it? Thanks for assistance in advance.
How would you count occurrences of a string (actually a char) within a string? I am doing something where I realised I wanted to count how many /s I could find in a string, and then it struck me, that there were several ways to do it, but couldn't decide on what the best (or easiest) was.  At the moment I'm going with something like:  string source = "/once/upon/a/time/"; int count = source.Length - source.Replace("/", "").Length;   But I don't like it at all, any takers?  I don't really want to dig out RegEx for this, do I?  I know my string is going to have the term I'm searching for, so you can assume that...  Of course for strings where length > 1,  string haystack = "/once/upon/a/time"; string needle = "/"; int needleCount = ( haystack.Length - haystack.Replace(needle,"").Length ) / needle.Length; Counterspelling a counterspell Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. How to get Beamer Math to look like Article Math In a LaTeX article, the math characters are more "curvy" than the text characters:  \documentclass[12pt]{article} \begin{document}  $x^2 + y^2 = z^2$ \end{document}     In a beamer presentation, the math characters do not have the same curves:  \documentclass{beamer} \begin{document} \begin{frame}  $x^2 + y^2 = z^2$ \end{frame} \end{document}     Question  How can I get the beamer math to look just like the article math?  I would prefer if the solution was a single line (that I could place at the top of my LaTeX file) as opposed to something that needs to appear next to every piece of math. A composite integer n is a Carmichael number if the only Fermat   witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not   coprime with $n$. The smallest example of such a number is $561 = 3 ·  11 · 17$   I think I understand the definition, but I don't quite understand why is the smallest Carmichael number is the above mentioned number. Please clarify. Create tar archive of a directory, except for hidden files? I'm wanting to create a tar archive of a specific directory (with its subdirectories of course). But when I do it, using the tar command, I get a list of files that were included, for example:     a calendar_final/._style.css      a calendar_final/style.css   As you can see, there are two versions of the same file. This goes for every file, and there are many. How do I exclude the temporary files, with the ._ prefix, from the tar archive? Why is quantum entanglement considered to be an active link between particles? “Airplane Mode” in OS X. Alternative command for `rfkill`? Mac OS X doesn't recognize rfkill from its terminal, and I am looking for an alternative command that has similar functionality (toggling "Airplane Mode").  NOTE: I don't wish to open an .app to toggle. I'd like to execute the command from command line or within script in Bash (or equivalents).    By "Airplane Mode", I meant to borrow the term from iOS support page . I would like to disable all incoming/outgoing network communications from my system. Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. I have a SCSI disk in a server (hardware Raid 1), 32G, ext3 filesytem. df tells me that the disk is 100% full. If I delete 1G this is correctly shown.  However, if I run a du -h -x / then du tells me that only 12G are used (I use -x because of some Samba mounts).  So my question is not about subtle differences between the du and df commands but about how I can find out what causes this huge difference?  I rebooted the machine for a fsck that went w/out errors. Should I run badblocks? lsof shows me no open deleted files, lost+found is empty and there is no obvious warn/err/fail statement in the messages file.  Feel free to ask for further details of the setup. So far, I understand that I have to look at each Jordan block. How do I prove that for a Jordan block with a value of x, $J_n(x)$, the Jordan decomposition of $(J_n(x))^2$ is $J_n(x^2)$ ? I have a feeling this will lead me to the solution. Thanks for your time. What is the exact difference between fluid and responsive design? What is the exact difference between fluid and responsive design?  I am bit confused about fluid and responsive design. Fluid is the one where we give width in percentages so that design will look good on browsers even if shrink or expand the browser. But what's different about responsive? The gluing lemma for closed sets states:     Given a finite closed cover $\{A_i\}$ of a topological space $X$, together with continuous maps $\{f_i : A_i \to Y\}$ into some other topological space $Y$, there exists a unique continuous map $f : X \to Y$ whose restriction to each $A_i$ is equal to $f_i$.   Question: What is a good and simple counter-example when the gluing lemma fails in the case that $\{A_i\}$ is infinite, but countably so.  My attempt: I have only been able find a silly counter-example: let $\{A_\alpha\}$ be the collection of all points of $X = [0,1] \subset \mathbb{R}$, and let $f_\alpha = 0$ for all $\alpha$ except for $\alpha = \alpha_0$, for which $A_{\alpha_0} = \{0\}$ and $f_{\alpha_0} = 1$. I want to append several shapefiles in QGIS. The ftools 0.6.1 add-on is installed.   But under Vektor>Datenmanagement-Werkzeuge>Shapedateien zu einer zusammenführen (=append shapefiles) is only the possiblility to choose the Eingabedatei (input file) and the Ausgabedatei (output file) but not a further Datei (file) for the append itself? (Even though there are  several shapefiles in the same folder)  //Furthermore the icon "In neue Projektion exportieren" (export to new projection) is also not appearing under the: Vektor>Datenmanagement-Werkzeuge? Do I need a further add-on?//  Has anyone an idea why I can't choose a shapefile for the appending?  Thanks in advance! phil Fix formatting errors of page number > 99 in table of contents? This is a follow up question of .  The formatting problem has nothing to do with cleanthesis.sty.  The problem is that if I use the charter font, page numbers greater than 99 are not formatted correctly in the table of content.  \documentclass[draft=on]{scrbook} \usepackage{blindtext} \usepackage{charter}  \listfiles  \begin{document} \tableofcontents \section{Section 1} \blindtext[100] \section{Section 2} \blindtext[500] \section{Section 3} \end{document} How do I get a slanted \sum symbol? I want to use a slanted \sum symbol denoting a different meaning from summation. How do I get a slanted \sum symbol? Texmaker generated pdf file not showing the position of cursor I am compiling LaTeX files with Texmaker on Ubuntu 12.04 system.   When I run pdflatex and then view the pdf file (internal), it does not show me the exact place where I am editing on Texmaker i.e. the generated pdf file does not show me the position of the cursor.   But some days before it was working fine. I don't know what causes this problem.  I already tried uninstalling and then reinstalling, I did update and upgrade too. But the problem still persists. How do I simulate in Photoshop what a piece will look like when printed on non-white paper? I'm designing a box in Photoshop that will be offset printed on unbleached paper. I haven't picked out a printer yet, so I don't know the exact kind of paper I'll be using and don't have access to it for proofs. For now I'm using a lightly-textured layer of tan color as a placeholder to mock it up.  What I'm trying to figure out is, how do I figure out how my design will look printed on natural paper rather than bleached white paper? I know that the brightness will be diminished and the hue will be altered, but I need to know how much so I can compensate for it as much as possible.  Is there a way to get at least a rough but reliably-simulated idea of how my prints will come out? Does it have to do with blend modes, or color separations? I'm at a bit of a loss here. How to speed up 4e combat I DM a group of four PC's and even "easy" battles take at least an hour to resolve. Normal encounters take even longer! I want to speed up combat so my players can do more in a session.  I've been trying a lot to speed up combat, we are already using:   Initiative cards that the players keep up 30 second bonus rule: if someone take his turn within 30 seconds, he receives a token. This token can be used to give a +1 on attack or damage, after rolling. Completely removed healing surges Checked all character sheets, they are roughly equal Roll damage and attack at the same time   I even introduced an experienced player, which helped my party to play more strategic, but speed hasn't improved.  Also the party is Lv 8, nearing Lv 9.  This is not an issue of the DM's turn taking too long. My turns take about 60 seconds tops. The 30 second rule helps a lot, most of the the players turns are within or near 30 seconds.   The major issue appears to be the sheer number of rounds, averaging on 25-30 rounds needed to resolve even a "normal" encounter.  What can I change to resolve combat faster?  I'm looking for both 4e solutions, system agnostic and home brew solutions. I've got `PDOException: SQLSTATE[HY000] [2002] No such file or directory` when using drush I installed Drush for the first time (so it may be a configuration error) and I'm trying to update to the latest core version of Drupal.  Drush is giving me this error - I'm on an OSX box.  dev5:clientnamedirectory my.name$ drush pm-update projects drupal-7.25 &lt;h1&gt;Additional uncaught exception thrown while handling exception.&lt;/h1&gt;  &lt;h2&gt;Original&lt;/h2&gt;&lt;p&gt;PDOException: SQLSTATE[HY000] [2002] No such file or directory in drupal_get_installed_schema_version() (line 155 of /Users/my.name/Sites/proface/includes/install.inc).&lt;/p&gt;  &lt;h2&gt;Additional&lt;/h2&gt;  &lt;p&gt;PDOException: SQLSTATE[HY000] [2002] No such file or directory in system_list() (line 165 of /Users/my.name/Sites/clientnamedirectory/includes/module.inc).&lt;/p&gt;&lt;hr /&gt;  Drush command terminated abnormally due to an unrecoverable error.