Proof of decreasing sequence
Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence
Basically, I've designed a booklet on Photoshop CC. It's an informational booklet so it's quite vibrant with lots of colours and pictures in it.   I've finished it and exported all the files into PDF format. Upon doing this, the PDF's are much "duller" for use of a better word. Now I'm not a designer and the whole thing was a learning process but now I'm reading about CMYK and RGB colour models and conversions etc. I designed the whole thing RGB (because CMYK looked duller obviously. The PDF's aren't disastrous, red is still red, black is still black but it's nowhere near as visually appealing anymore.   Is this just because of the monitor and the models each program uses or when I send my brochure to be printed, am I doomed for a much duller booklet? There's obviously a way around it otherwise all print would be dull. I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. How do I restore the sources.list file? I accidentally removed my /etc/apt/sources.list file. Is it hosted by Canonical somewhere so I can re-add it to the system? It's a bit of an emergency, so I hope someone can provide a quick answer. I can't figure out how to line up the symbols of an itemize environment in beamer with the left part of the main text. In other words, I would like to supress the left margin in beamer itemize lists.  It would be nice if the solution worked with any itemize label defined with  \setbeamertemplate{itemize item}[]   In order to make myself clearer, what beamer produces by default is something like:  Main text. Main text. Main text...    * Item 1    * Item 2    * Item 3    ....   and the desired result is   Main text. Main text. Main text... * Item 1 * Item 2 * Item 3 .... Omitting article “the” in front of plural nouns I'm helping someone learn English (second language), and I'm having trouble explaining when to use the and when not to. For example:     Students learn better when teachers are passionate about teaching.   versus     The students learn better when the teachers are passionate about teaching.   There's a subtle difference. The first one is a general statement, whereas the second might imply a study where a group of students learned better than another.  Is the difference simply that, general v. specific? How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? Benevolent aliens who (initially) refuse to show themselves This is a sci-fi book where aliens arrive on Earth and push Earth towards world peace, but refuse to reveal themselves physically. When the aliens finally do show themselves, we find that they       look like .   I recall that the aliens seemed benevolent, but that humanity did not like the direction they were being told to go. I cannot remember the ending or if the aliens turned out to be truly benevolent. Why would you not want to rollover a previous employer's 401(k) when changing jobs? Are there any valid reasons to not rollover a former 401(k) when changing jobs?  I have several little ones that I never quite got around to rolling together - with an upcoming job change, should they all be combined into my new employer's retirement plan?  If not, what else might be considered? Attributive nouns vs. of-genitive How can I run an external program from C and parse its output? I've got a utility that outputs a list of files required by a game.  How can I run that utility within a C program and grab its output so I can act on it within the same program? UPDATE: Good call on the lack of information.  The utility spits out a series of strings, and this is supposed to be portable across Mac/Windows/Linux.  Please note, I'm looking for a programmatic way to execute the utility and retain its output (which goes to stdout). So, today is July 21st and . gives me an option to download 16.04.1 LTS. However, attempting to upgrade gets me this message -  $ sudo do-release-upgrade Checking for a new Ubuntu release No new release found   What is wrong? Does it matter that I am running apt-get update behind apt-cacher-ng?  Note: Commenting out the files in apt.conf.d that used apt-cacher-ng did not make a difference.  Edit: Explanation for possible duplicates:  deals with upgrade from 12.04LTS to 14.04LTS. My question is specific to 14.04 LTS to 16.04.1 LTS upgrade. Old answers, while good for archival purposes are less relevant in a fast changing package like ubuntu. Moreover none of the answers answer what is wrong. The link  pointed out is mentioned in one of the comments. This Q &amp; A is more helpful for people upgrading today than the work around mentioned in "". Your summary page shows "N Edits to your posts" for the selected time period but you really have to be looking for it.  Could we add edits to the notification feed? You could put the reason for the edit (entered--or not--when the editor changes it) and just a link to the post. Much as I loathe Microsquish's Word, it does allow you to insert into a header or footer the automated text "Page X of Y."   When using InDesign, it's easy to set up the "Page X" part on a master page. But is there any placeholder character for the Y? That is, it will count the number of pages in the document, either in an absolute or a relative-to-the-section fashion? Every now and again I narrowly miss sending out something with a footer which reads "Page 4 of 3," and I would love to know if there's some way around that. How to remove images from showing in a post with the_content()? Trying to learn more about creating my own themes and not using any plugins I wanted an ability to display all the images in a post excluding the thumbnail in my single-foobar.php file and I can do that with:  &lt;?php if ( have_posts() ) : while ( have_posts() ) : the_post();      $thumb_ID = get_post_thumbnail_id( $post-&gt;ID );  $args = array(    'post_type' =&gt; 'attachment',    'numberposts' =&gt; -1,    'post_status' =&gt; null,    'post_parent' =&gt; $post-&gt;ID,    'exclude' =&gt; $thumb_ID   );  $attachments = get_posts( $args );  if ( $attachments ) {     foreach ( $attachments as $attachment ) {        echo '&lt;div class="portproject"&gt;';        $image_res = wp_get_attachment_image( $attachment-&gt;ID, ' img-responsive' );        echo $image_res;        echo '&lt;/div&gt;';       }  }  else { ?&gt;     &lt;span&gt; No pictures loaded at this time&lt;/span&gt;  &lt;?php } endwhile; endif; ?&gt;   However, when I apply my static HTML and then try to call get_the_content() with:  &lt;?php  $content = get_the_content(); $content = preg_replace("/&lt;img[^&gt;]+\&gt;/i", "(image) ", $content);           $content = apply_filters('the_content', $content); $content = str_replace(']]&gt;', ']]&gt;', $content); echo $content; ?&gt;   it removes the images but leaves a (image) in the post with spaces then displays the remaining text. I did find a similar  but it suggests using get_the_content() but that strips all the styling from the post.  So my question is how can I strip images from the_content()?  I did find "" from my searches but that still uses the get_the_content() approach. solving diophantine equations using congruences So, a question in Elementary Number Theory by David M. Burton:     Problems 4.4      2(a) Using congruences, solve the diophantine equation 4x + 51y = 9       Hint: $4x \equiv 9(mod \ 51) $ gives $ x = 15 +51t$, whereas $ \ 51y \equiv 9 (mod \ 4)$ gives $y = 3+4s $. Find the relation between $s$ and $t$ From this hint, I substitute $ x = 15 +51t$ and $y = 3+4s$ in the given equation $4x + 51y = 9$ to get the relation $s = -t-1$ and using which I can get the solutions $ x = 15 + 51t, \ y =-1-4t$ But we know that the solutions of $ax \equiv b (mod \ n)$ is equivatent to the solutions of the diophantine equation $ax -ny = b$, so, solutions of $4x \equiv 9(mod \ 51) $ is equivalent to solutions of the equation $4x-51y=9$ whereas, solutions of $51y \equiv 9(mod \ 4) $ is equivalent to solutions of the equation $-4x+51y=9$ Now, my question is: as, the above equations $4x-51y=9$ and $-4x+51y=9$ are different, then what is the point of substituting $ x = 15 +51t$ (solution of $4x-51y=9$) and $y = 3+4s $ (solution of $-4x+51y=9$) in the given equation $4x+51y=9$ to find a relation between $t$ and $s$ ? Isn't it absurd as $ x = 15 +51t$ and $y = 3+4s $ are solutions of two different equations? Can anyone please explain this? PS: I did it like this $4x \equiv 9(mod \ 51) $ gives $ x = 15 +51t$ and by substituting it in $4x+51y=9$ we get $y=-1-4t$ Where is the itoa function in Linux? itoa() is a really handy function to convert a number to a string. Linux does not seem to have itoa(), is there an equivalent function or do I have to use sprintf(str, "%d", num)? With "magic quotes" disabled, why does PHP/WordPress continue to auto-escape my POST data? What does 5' and 3' mean in DNA and RNA strands? What are 5' and 3' in DNA and RNA strands? Please clarify with some images and please use simple English. Kill process after it's been allowed to run for some time I want to limit the time a grep process command is allowed to run or be alive.  For example. I want to perform the following:  grep -qsRw -m1 "parameter" /var   But before running the grep command I want to limit how long the grep process is to live, say no longer than 30 seconds.  How do I do this?  And if it can, how do I return or reset to have no time limit afterwards. GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light