Show $0< \frac{2x}{\pi} < \sin x$ How do I go about showing $0 \leq \frac{2x}{\pi} \leq \sin x $ for $x\in[0,\pi/2]$?  I am completely stuck where to start.  Many thanks.  (I see it is a step in the proof of Jordan's lemma, but I'm not interested in this, and the proofs I find do not explain this actual step, ta).
How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$? I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear $f'(x)$ is decreasing and has a zero within the domain, so I'm not sure how this might be used to prove the inequality.
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. Installing Wi-Fi driver for Realtek Semiconductor RTL8723DE Device [10ec:d723] How do I synchronise two folders in real-time in Windows 7? I want to synchronise two folders in real time under Windows 7.  Basically, I want to monitor a folder and synchronise each change (new files, changed files, deleted files) to another drive. It has to be in real time, so it gets synchronised instantly when a change happens. A one-direction synchronisation is enough.  I tried Microsoft's , but it does only syncing by hand or scheduled (thus not satisfying the "real-time" requirement).  Can this be achieved with Windows 7 itself? If not, are there other methods of doing this? When my phone MOTOROLA G3(2015) is updated to Marshmallow, its not showing files when connected to PC. While connecting, it shows options as charing only, PTP, MTP, MIDI. When I select MTP, it shows the external card, but not showing the contents, where as internal memory is detected and all files are shown. Why is my Wi-Fi connection speed much slower than a wired connection? When I plug my laptop via LAN cable and check speed on speedtest.net I have over 32 Mb/sec. When I unplug the cable and connect to the same router I have less than 15 Mb/sec. I tried to change channels already but this didn't help. I'm sitting just near the router.   What can it be? The router is a Dlink 400. Titles of articles I'm about to cite contain upper case letters and when using BibTeX it converts them to lower case ones. This happens only in the title and only the first letter conserves its case. For example, when I cite an article about HF, the reader won't know if it is about Hafnium (Hf) or fluorine acid (HF).  I know that I can fix it manually in the .bbl file but I would like to avoid it, or fix it automatically. Let's say I use a  with a lot of code inside. To improve readability and reusability I'd like to use helper/private methods. However, a Drupal .module file is not a class.  The idea is to split big methods from this file into more readable and reusable smaller methods.  What is the correct way to call helper/private methods from a Drupal .module file? Fixed background image with ios7 Difference of the \dots* How secure would github hosting be for private repositories? How do suggested edits work? We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact,  $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$$  How do we prove that, more generally, for $p &gt; 1$ then,  $$\sum_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$$  (The above simply was the case $p = 10$.) Long sites break the layout on the front page The long name pushes the stats to a weird place. And what's with the "Stack Exchange" at the end anyway? "This product is good value" vs "This product is of good value" in a sentence 1.This product is good value. 2.This product is of good value.    Are there any difference in meaning between "good value" and "of good value"? Getting around the FAT32 4GB file size limit In PHP (>= 5.0), is passing by reference faster? In PHP, function parameters can be passed by reference by prepending an ampersand to the parameter in the function declaration, like so:  function foo(&amp;$bar) {     // ... }   Now, I am aware that this is not designed to improve performance, but to allow functions to change variables that are normally out of their scope.  Instead, PHP seems to use Copy On Write to avoid copying objects (and maybe also arrays) until they are changed. So, for functions that do not change their parameters, the effect should be the same as if you had passed them by reference.  However, I was wondering if the Copy On Write logic maybe is shortcircuited on pass-by-reference and whether that has any performance impact.  ETA: To be sure, I assume that it's not faster, and I am well aware that this is not what references are for. So I think my own guesses are quite good, I'm just looking for an answer from someone who really knows what's definitely happening under the hood. In five years of PHP development, I've always found it hard to get quality information on PHP internals short from reading the source. Reference - What does this error mean in PHP? I've lost my portal home, and I am invincible. How can I leave the nether? I've gone into the Nether and seemed to have lost my portal out of it, I also can't die in lava or by mobs attacking me, I'm appear to be invincible and can't die.  This is Single Player by the way.  Why is this happening? What is the maximum number of pucture repairs an inner tube can take? Simple enough question: is there some shell command (or GUI method) I can use that, given the path to a file on my system, tells me what package put it there? Assuming the file did in fact come from a package, that is.  Bonus question: what if it's a file that isn't installed on my system? Is there, say, a website that will let me look up a file and see what packages, if any, provide it?