If $s_1=\sqrt 2$, and $$s_{n+1}=\sqrt {2+\sqrt{s_n}}$$  Prove that ${s_n}$ is converging, and that $s_n&lt;2$ for all $n=1,2\ldots$  My attempt to use mathematical induction: if $s_n&lt;2$ then I use $s_{n+1}=\sqrt {2+\sqrt s_n}$, to prove $s_{n+1}&lt;\sqrt {2+2}=2$. I guess $s_n$ is an increasing sequence, I can use the $s_n$ is bounded and increasing, and then prove that $s_n$ converges.
Here's Prob. 3, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:      If $s_1 = \sqrt{2}$, and $$s_{n+1} = \sqrt{2 + \sqrt{s_n}} \ \ (n = 1, 2, 3, \ldots),$$ prove that $\left\{ s_n \right\}$ converges, and that $s_n &lt; 2$ for $n = 1, 2, 3, \ldots$.    My effort:   We can show that $\sqrt{2} \leq s_n \leq 2$ for all $n = 1, 2, 3, \ldots$. [Am I right?]  Then we can also show that $s_n &lt; s_{n+1}$ for all $n = 1, 2, 3, \ldots$. [Am I right?]  But how to calculate the exact value of the limit? Where does this sequence occur in applications?
I am an Indian student in France on a 6 month visa for an Internship. Currently I don't have my passport with me as I have applied for UK visa and am not hoping to receive it back in next 10 days. I was wondering if I can travel to Germany without my passport in hand.   The documents I can carry are a photocopy of my passport's identity page, Residence proof here, proof of Internship in France,Indian national identity card, receipt from UK embassy stating my visa application.   Please note that I do NOT have a photocopy of French visa page. Wrongly set chmod / 777. Problems? I was trying to run chmod -R 777 ./ but ended up typing chmod -R 777 / and set 777 on my entire machine. What can go wrong? How can I fix it? How to quickly and safely defrost chicken? Two diagrams side by side I am trying to put two feynman diagrams side by side, but they get displayed below eachother. I am using feynmf and subfloat. This normally works for pictures.  \begin{figure}[ht!]       \centering     \subfloat{          \begin{fmfgraph*}(40, 25)         \fmfstraight          \fmfbottom{wm,i2,z2}           \fmftop{wp,i1,z1}           \fmf{dashes}{wp,i1,z1}           \fmf{dashes}{z2,i2,wm}           \fmf{dashes,caption=$W^{+}$}{i1,i2}         %  \fmf{dashes,caption=$Z^0$}{i2,z1}          % \fmf{dashes,caption=$Z^0$}{i2,z2}         \fmfv{lab=$Z^0$}{z1}         \fmfv{lab=$Z^0$}{z2}     \fmfv{lab=$W^+$}{wp}     \fmfv{lab=$W^-$}{wm}         \end{fmfgraph*}     }      \subfloat{          \begin{fmfgraph*}(40, 25)         \fmfstraight          \fmfbottom{wm,i2,z2}           \fmftop{wp,i1,z1}           \fmf{dashes}{wp,i1,z1}           \fmf{dashes}{z2,i2,wm}           \fmf{dashes,caption=$W^{+}$}{i1,i2}         %  \fmf{dashes,caption=$Z^0$}{i2,z1}          % \fmf{dashes,caption=$Z^0$}{i2,z2}         \fmfv{lab=$Z^0$}{z1}         \fmfv{lab=$Z^0$}{z2}     \fmfv{lab=$W^+$}{wp}     \fmfv{lab=$W^-$}{wm}         \end{fmfgraph*}     }      \caption{Exchange of W boson}     \end{figure} How to convert an instance of std::string to lower case I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time? Something along the lines of 'hero of the occasion', but specifically for birthday? If there isn't, how would you otherwise say that?  ('the subject of birthday party', 'the hero of this birthday party', 'birthday's hero' all sound stupid somehow). How can I install Steam? Say we have the following sample and we are trying to estimate the variance of the sample mean of the population.  X = [0, -1, 2, 10, -3]   If I take an increasing number of bootstrap (e.g. 100, 1000, 10,000, 100,0000), the bootstrap distribution of the sample mean gets narrower and narrower. In other words, I can make the bootstrap distribution arbitrarily narrow.   Given the above, what's the point of using a different number of bootstrap samples here? And wouldn't this mean that I could make the standard error of the mean arbitrarily small by just taking more and more bootstrap samples? What am I missing?  Please note that I am doing a proper bootstrap sample in the sense that each bootstrap sample has the same exact size as the original sample (input array). Are the complements of two homeomorphic compact, connected subsets of $\mathbb{R}^2$ homeomorphic? I want to copy any file to my external hard drive. But always when I tried to do this I have next output:  cp: cannot create regular file /media/NameOfCard/[file]' : Read only file system  I also tried:  sudo chmod -R 775 /media/NameOfCard sudo chmod -R 777 /media/NameOfCard   But the same effect - ... Read only file system  Update:   After mount I have next output:  nazar_art@nazar-desctop:~$ mount /dev/sda1 on / type ext4 (rw,errors=remount-ro) proc on /proc type proc (rw,noexec,nosuid,nodev) sysfs on /sys type sysfs (rw,noexec,nosuid,nodev) none on /sys/fs/fuse/connections type fusectl (rw) none on /sys/kernel/debug type debugfs (rw) none on /sys/kernel/security type securityfs (rw) udev on /dev type devtmpfs (rw,mode=0755) devpts on /dev/pts type devpts (rw,noexec,nosuid,gid=5,mode=0620) tmpfs on /run type tmpfs (rw,noexec,nosuid,size=10%,mode=0755) none on /run/lock type tmpfs (rw,noexec,nosuid,nodev,size=5242880) none on /run/shm type tmpfs (rw,nosuid,nodev) /dev/sdb1 on /media/sdb1 type vfat (rw) binfmt_misc on /proc/sys/fs/binfmt_misc type binfmt_misc (rw,noexec,nosuid,nodev) gvfs-fuse-daemon on /home/nazar_art/.gvfs type fuse.gvfs-fuse-daemon (rw,nosuid,nodev,user=nazar_art)   Any suggestions?   How to solve this trouble and let permission to copy and delete files to hard drive? How do you find the modular inverse of $5\pmod{\!11}$ Show that $E(XE[X|G])=E[(E[X|G)^2]$ How do get my sitelinks to appear on Google search results? Can I revert to the Mountain Lion style labels instead of Mavericks tags? How? Can command substitution be nested in variable substitution? When I try to create a reaction from chemmacros/reactions I got the following error. I dunno what I am missing. Please help. Thank you.     line 11: Undefined control sequence. \end   line 11: Missing number, treated as zero. \end   line 11: Missing number, treated as zero. \end     Minimum Working Example    \documentclass[12pt,a4paper]{article} \usepackage{chemmacros} \chemsetup{modules={reactions}} \begin{document} \begin{reaction}     A + B -&gt; C \end{reaction} \end{document} There are two syntaxes for command substitution: with dollar-parentheses and with backticks. Running top -p $(pidof init) and top -p `pidof init` gives the same output. Are these two ways of doing the same thing, or are there differences? mknod /tmp/oracle.pipe p  sqlplus / as sysdba &lt;&lt; _EOF set escape on  host nohup gzip -c &lt; /tmp/oracle.pipe &gt; /tmp/out1.gz \&amp; spool /tmp/oracle.pipe select * from employee; spool off  _EOF  rm /tmp/oracle.pip   I need to insert a trailer at the end of the zipped file out1.gz , I can count the lines using  count=zcat out1.gz |wc -l   How do i insert the trailer  T5 (assuming count=5)   At the end of out1.gz without unzipping it. In ideal gas model,  is the measure of average kinetic energy of the gas molecules. If by some means the gas particles are accelerated to a very high speed in one direction, KE certainly increased, can we say the gas becomes hotter? Do we need to distinguish the random vibration KE and KE in one direction?  Furthermore, if we accelerate a block of metal with ultrasonic vibrator so that the metal is vibrating in very high speed with cyclic motion, can we say the metal is hot when it is moving but suddenly become much cooler when the vibration stop? Image capture opens after iPhone is connected Everytime I connect my iPhone to my laptop, Image Capture opens. How to I stop it? I just need to use iTunes, not Image Capture.