For $c &gt; 1$, define sequence $\{a_n\}$ with $a_1 = \sqrt{c(c-1)}$ and $a_{n+1} = \sqrt{c(c-1) + a_n}$, $n &gt; 1$. Prove that $\{a_n\}$ is a monotonically increasing   sequence and it's bounded by $c$.   My attempt:  Obviously $a_1 &lt; a_2$, by induction, suppose $a_n &gt; a_{n-1}$, i.e. $a_n - a_{n-1} &gt; 0$. We have $a_{n+1}^2 - a_n^2 = c(c-1) + a_n - c(c-1) - a_{n-1} = a_n - a_{n-1} &gt; 0$. Hence, $\{a_n\}$ is a monotonically increasing sequence. Otherwise, $a_1 = \sqrt{c(c-1)} \leq c$ because $c^2 - a_1^2 = c^2 - c(c-1) = c \geq 0$, by induction, suppose $a_n \leq c$, we prove that $a_{n+1} \leq c$. $a_{n+1} = \sqrt{c(c-1) + a_n} &lt; \sqrt{c(c-1) + c}$, but $c^2 - \left(\sqrt{c(c-1) + c}\right)^2 \geq 0$, then $a_{n+1} \leq c$.  Is this solution true? Thank all!
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
changing notification emails from WordPress @mydomain.net to something else How do I change notification emails address from WordPress @mydomain.net to something else.  I want to do this because WordPress @mydomain.net ends up getting flagged as junk mail.  Thanks  Daniel Can the Next/Prev Post links be ordered by menu order or by a meta key? I have a series of posts that are ordered by a meta_key value. They could also be arranged by menu order, if necessary.  The next/prev post links (generated by next_post_link, previous_post_link, or posts_nav_link all navigate by chronology. While I understand this default behaviour, I don't understand how to change it. I found that it maps through to adjacent_post_link in link-template.php, but then it starts to seem fairly hard-coded. Is it recommended to re-write this from scratch to replace it, or is there a better solution. If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$ Let $f:[0,1]\to \mathbb{R}$ is continuous, such that $f(0)=f(1)$. For each $n\in\mathbb{N}$, prove that $\exists x\in [0,1]$ such that $x+\dfrac{1}{n}\in[0,1]$ and $f\left(x+\dfrac{1}{n}\right)=f(x)$.   My approach: The interval $[0,1]$ is connected, then if $f$ is continuous, we have $f([0,1])\subset\mathbb{R}$ is a interval. And I suppose that, I can define a closed path such that $f(0)=f(1)$. Any hint pls, Typesetting UML class diagrams What's the best way to draw UML class diagrams in LaTeX? Disable back button in android What to conclude from this lasso plot (glmnet) Following is the plot of glmnet with default alpha(1, hence lasso) using mtcars data set in R with mpg as the DV and others as the predictor variables.  glmnet(as.matrix(mtcars[-1]), mtcars[,1])     What can we conclude from this plot regarding different variables, especially am, cyl and wt (red, black and light blue lines)? How would we phrase the output in a report to be published?  I thought of following:   wt is the most important predictor of mpg. It is negatively affecting the mpg. cyl is a weak negative predictor of mpg. am may be a positive predictor of mpg. Other variables are not robust predictors of mpg.    Thanks for your thoughts on this.  (Note: cyl is the black line which does not reach 0 till very close to it.)  Edit: Following is plot(mod, xvar='lambda') which shows x-axis in order reverse of above plot: Python - email header decoding UTF-8 Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$.   I am reasoning for the absurd and I am assuming that $gcd (x ^ 5, x ^ 6) = ax ^ 2 + bx ^ 3\in\mathbb{Q}[x^2 ,x^3 ] $, then, for the Bezout identity there exists $p (x) = cx ^ 2 + dx^3$ and  $q(x)= ex ^ 2 + fx ^ 3$ in $\mathbb{Q}[x^2 ,x^3 ]$ such that $x ^ 5p (x) + x ^ 6q (x) = ax ^ 2 + bx ^ 3$, which is absurd since $$9 = deg (x ^ 5p (x) + x ^ 6q (x)) \neq  deg (ax ^ 2 + bx ^ 3) = 3.$$  Is it okay what I did? Thank you very much. I have a virtual machine running ubuntu and I am trying to host some files on ubuntu and map the drive in windows PC.  Following the steps , I am able to see (from windows) the "work" folder after typing in \ubuntu_ip. But when trying to access or map it with "svc_acct" from windows, I get "0x80070043 error the network name cannot be found". I guess I am not accessing correctly from windows.   SAMBA config below (svc_acct is an account set up under linux). When accessing the folder from windows, I get prompted for username/password. Not sure what credential to type in exactly.  samba config: [work] path = /home/svc_acct/work valid users = svc_acct available = yes read only = no browsable = yes public = yes writable = yes force user = svc_acct create mask = 0777 directory mask = 0777   UPDATE:  All I did to resolve this is by changing samba config's path from path = /home/svc_acct/work to path = /home/svc_acct following  post  (last sentence). Then no more prompt for password in windows when I try to click into the folder. Thanks to @sancho.s for suggesting the link. I just can't seem to figure this one out.  I want to use the command line to connect to a MySQL database residing on another server.  I went ahead and created the username and password for the user.  I have also granted all privileges on that user for that database.  When using the command: mysql -h &lt;hostname&gt; -u &lt;username&gt; -p, I get the following error:  ERROR 1045 (28000): Access denied for user '&lt;username&gt;'@'&lt;local_machine_hostname&gt;' (using password: YES)   The problem is that it keeps appending the current machine's hostname into the username.  Obviously, that user@&lt;local_machine_hostname&gt; is not correct.  It doesn't matter what I type.  For instance, if I type:  mysql -h &lt;hostname&gt; -u '&lt;username&gt;'@'&lt;hostname&gt;' -p   It does the same, only in the error output, it says:  Access denied for user '&lt;username&gt;@&lt;hostname&gt;'@'&lt;local_machine_hostname&gt;'   Is there a setting in a configuration file which is allowing this to happen?  It's really quite annoying.  I need to set up a tikiwiki server, and it cannot connect because during the step where you set up mysql, it keeps appending the local machine's hostname to the mysql login name. So this homework question is in the context of $\mathbb{R}$ only, and we are using Lebesgue measure.  The sum $A+B$ is defined to be $A+B=\{a+b|a\in A,b\in B\}$. The question is: If $m(A),m(B)&gt;0$, is it necessarily the case that $A+B$ contains an open interval?  This question is quite annoying, since it is quite difficult to calculate the sum even after you construct an example. However, it seems like no matter how I constructed a set of positive measure, there is a region that have a lot of point in it (yet this region is not necessarily dense). This suggest that when summed up, the holes should be covered up somehow and thus there would be an open interval, and so the answer is yes. Sorry for a rather vague idea, but I'm just not sure what to do with this problem. I have a feeling that somehow topology would have a way to capture precisely the idea above about a region with a lot of point.  (also, I think my middle school teacher mentioned this once, and I remembered the answer to be yes; but that is quite a while ago though)  Thank you. Any help will be appreciated.  EDIT2: So Camilo's answer is gone. I am looking at the convolution method right now. Because the convolution is continuous, if it is nonzero I am done. But then there is no guarantee that the convolution is nonzero even if $x\in A+B$. I still need help with this direction. Can someone give me some hints?  EDIT: Camilo's answer below essentially suggest the method of getting 2 open interval from regularity theorem, one cover a subset of A and one cover a subset of B such that the sum of the measure of these two subset is bigger than the measure of the larger interval. user99680's answer point to a wiki which show a method that essentially simply acquire an open set. After some thinking, I believe that these method simply can't be made to work. I constructed this example in the hope that it would convince you that it is the case:  Let $I_{0},I_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is nonzero and $m(I_{n})=9^{-n}$ and all of them are inside $[0,2]$. Let $J_{0},J_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is at least $3$ and $m(J_{3n})=m(J_{3n+1})=m(J_{3n+2})=\frac{m(I_{n})}{3}$. Let $O=\bigcup I_{n}$ and $H=\bigcup J_{n}$. Construct $A\subset O$ such that $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$. Construct $B\subset H$ such that $m(B\bigcap J_{n})=\frac{3m(J_{n})}{4}$. Now if you apply Camilo's method, it is possible that your $I$ would be one of the $I_{n}$ above, and your $J$ is one of the $J_{m}$ above. No matter which $I_{n},J_{m}$ you have, the argument does not work. Rewrite $m=3q+r$ where $0\leq r\leq 2$. If $q&lt;n$ then $m(J_{m})&gt;m(I_{n})$ so the longer interval have length $\frac{m(I_{q})}{3}$; $m(A\bigcap I_{n})$ is at most $\frac{3m(I_{q+1})}{4}=\frac{m(I_{q})}{12}$ and $m(B\bigcap J_{m})=\frac{3m(J_{m})}{4}=\frac{m(I_{q})}{4}$. Thus $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. If $n\geq q$ then $m(I_{n})&gt;m(J_{m})$ so the larger interval is of length $m(I_{n})$; $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$ and $m(B\bigcap J_{m})$ is at most $\frac{3m(J_{3n})}{4}=\frac{m(I_{n})}{4}$. Once again $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. Hence the Camilo's argument does not work. user99680's method does not work either, since you get $O$ and $H$ which is even bigger than any interval. Assessing approximate distribution of data based on a histogram Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right).   Depending on how I group or bin the data, I can get wildly different histograms.   One set of histograms will make is seem that the data is exponential. Another set will make it seem that data are not exponential. How do I make determining distributions from histograms well defined? How do I figure the dice and bonuses for attack rolls and damage rolls? I'm building my character, a Half-Elf Ranger, and I'm a little confused about how to work out the attack bonus and the damage bonus.  So, please tell me if I've got this right:   I'm proficient with a longbow. That means my attack roll is 1d20 + prof bonus + Dex modifier. The damage roll however is 1d8 + Dex modifier.   I'm just a little confused about working out the numbers (without even going into spells and when to roll for attack, etc.) An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$ Converting ISO 8601-compliant String to java.util.Date In setting up dual Boot with Windows and Ubuntu, which OS do I install first? I'd like to install both Ubuntu 12.04 and Windows XP on a Dell laptop, and I was thinking about using a dual boot structure, and using the bulk of my hard drive as empty hard drive space to share files between the two operating systems (so choice of file system type is very important in this set-up).     The kind of partitioning structure I would like to use is  Partition 1 - Ubuntu 12.04 (root) (20GB)  Partition 2 - Ubuntu /home        (20GB)  Partition 3 : Free Space          (560GB)  Partition 4 : Windows XP          (35GB)  Partition 5 : SWAP                (3GB)  (Total Hardrive Capacity is ~640GB)    My question is; what is the best way to go about setting up this kind this system?   Should I install Windows XP first and setup the partitions, and then install Ubuntu which I believe will install the GRUB bootloader for OS booting choice   or   Do I install Ubuntu first, setting up the available partitions and then perform a WIndows install?  Please let me know if there is anything in this setup that I have left out and should know about, including things related to setting particular partitions as logical or primary, and whether the boot partition and the filesystem partition should actually be two separate partitions. Using "nor" in a list without "neither" This is the classic usage of neither/nor:     I would neither hide nor run away.   But is the following construction grammatical? (More than two choices, no neither)     I wasn't going to play dead.   I wasn't going to hide.   Nor was I going to run away. Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? Difference between sentences using 'was' and 'being'? What is the meaning of below sentences using was and being?     He committed suicide after he was severely beaten up by four teachers. He committed suicide after being severely beaten up by four teachers. Gravity as a gauge theory Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory).   Why should "classical" gravity be some (non-trivial or "special" or extended) gauge theory? Should quantum gravity be a gauge theory?  Remark: There are some contradictory claims in the literature to this issue. To what extent gravity is "a gauge" theory?Obviously, GR is not a YM theory. So, why do some people say that gravity "IS" a gauge theory? I found this question relevant, e.g., then we handle GR in the Einstein-Cartan theory or any other theory beyond GR, like teleparallel theories or higher-derivative gravitational theories. So I think it could be useful to discuss the "gauge flavor" of gravity here.