Convergence of a sequence defined by a recurrence inequality Let $(u_n)$ be a sequence of real numbers with the following property \begin{eqnarray*} \forall\,m,n\ge1,\quad u_{m+n}\le \frac{m}{m+n}u_m+\frac{n}{m+n}u_n \end{eqnarray*} Is it true that $(u_n)$ converges to $\inf u_n$ which is either $-\infty$ or a finite $u\in\mathbb{R}$? There's no obvious monotonicity property that I was able to use. It's not hard to see that $u_{2n}\le u_n$, but I was unable to make any use of it. My attempts to show that $(u_n)$ is Cauchy were not successful. Neither was my attempt to prove that $(u_{2n})$ and $(u_{2n+1})$ are convergent. Any hints would be appreciated.
Subadditive sequences and the limit $\sup_{n\to\infty} a_n / n$ Let $\{ a_n, n\geq 1\}$ be a sub-additive sequence, i.e. for any $m,p\geq 1$ $$ a_{m+p} \leq a_m + a_p. $$ .Prove that $$\lim_{n\to\infty} \frac{a_n}{n} = \inf_n \frac{a_n}{n}.$$ I guess the first step is to state $$a_{n} = a_{mp+q} \leq a_{mp} + a_q$$ but I'm not sure where to go from here....
How can I communicate with a Unix domain socket via the shell on Debian Squeeze? I’m running a Debian Squeeze web server. I’ve installed memcached on it, and configured memcached to listen on a Unix domain socket (at /tmp/memcached.sock), as it only needs to receive messages from the website, which lives on the same server.  It seems to be working fine, but I’d also like to communicate with memcached via the shell, to check that it’s doing what I think it’s doing.  memcached accepts messages via a simple ASCII protocol (if I understand correctly). If it was listening on TCP/IP, I could send messages to it via e.g. nc:  $ echo "stats settings" | nc localhost 11211   But I can’t figure out how to send that text to the domain socket instead.  On my laptop (which runs OS X Lion), both nc and telnet have options (-U and -u respectively) to use domain sockets. However, on my Debian Squeeze web server, these options aren’t present. Ubuntu Software doesn't open in Ubuntu 20.04 How do you access the Shrine of Winter? In Dark Souls 2 there is a 'Shrine of Winter' near Sinners Rise. When you approach the shrine it will not let you open the door and a message pops up that says      Seek Mightier Souls   What does this mean and how do you open this shrine? Gdata signed request error: "Unknown authorization header" Do you get to use the additional attack from the Extra Attack feature as well when you Ready an Attack action?  One of my players, a monk, decided to Ready an action as an earth elemental was about to attack. The trigger was the elemental moving within five feet of him. When this occurred, he took his Attack action, and then proceeded to take the extra attack as well.  I was unsure at the time whether or not the extra attack would also happen, and I haven't had much luck figuring it out. Replacing all values in a table > $T$ I'm curious why PCBs have one square pad in row for 0.1" headers. What is purpose of putting one square pad? What it indicates? Can you push more amps through square pad than round? Should you use a plural verb if you refer to a fraction?     Nine tenths of the pillar have rotted away.   I think has should be used here, but I'm not sure. I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?  Consider this:  set.seed(999) prcomp(data.frame(1:10,rnorm(10)))$x              PC1        PC2  [1,] -4.508620 -0.2567655  [2,] -3.373772 -1.1369417  [3,] -2.679669  1.0903445  [4,] -1.615837  0.7108631  [5,] -0.548879  0.3093389  [6,]  0.481756  0.1639112  [7,]  1.656178 -0.9952875  [8,]  2.560345 -0.2490548  [9,]  3.508442  0.1874520 [10,]  4.520055  0.1761397  set.seed(999) princomp(data.frame(1:10,rnorm(10)))$scores          Comp.1     Comp.2  [1,]  4.508620  0.2567655  [2,]  3.373772  1.1369417  [3,]  2.679669 -1.0903445  [4,]  1.615837 -0.7108631  [5,]  0.548879 -0.3093389  [6,] -0.481756 -0.1639112  [7,] -1.656178  0.9952875  [8,] -2.560345  0.2490548  [9,] -3.508442 -0.1874520 [10,] -4.520055 -0.1761397   Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?    In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)? Why is a nonstop flight sometimes more expensive than multiple connections? Usually I find a flight I like that uses one or more connections with the same or different companies. In these cases, if the distance between the first and the second connection is not too large, I prefer to go by bus or train and then take the flight. The problem is that sometimes it is more expensive to take the flight in the second airport than to take the flight from the first one. I mean, I'm wondering why sometimes it is more expensive to take only a flight than to take the same flight and another one.  A practical example (and real, you can check with SkyScanner). Imagine I am in Madrid (Spain) and would like to go to Rio de Janeiro (Brazil) on April 1st. The cheapest flight I have found is this:     TAM airlines: MAD -> GRU -> GIG : 507 €   Fantastic! GRU is one of São Paulo's airports. That's perfect for me! I'll go on to Rio traveling overland. But then, I look for this flight (MAD -> GRU, non-stop):     TAM airlines: MAD -> GRU : 597 €   It's the same flight, 90 € more expensive! I've found this kind of behavior a lot of times and I'm intrigued to know why it happens. Order and degree of a differential equation How do you handle uneven tiles while rendering a tile map? Here is what I want to do with my tile map that I am unsure of. As you can see the top walls are way larger then the bottom and side ones (this is also an issue with my corners which are odd shapes [more like an L then a square] as well as larger then the 40x40px that my current tiles are).    I have also attached the tile sheet I am using for the tiles to show you what the corners and tops look like compared to the rest of the tiles.  What I’m thinking is I might be able to just draw them using the same array I have if I use the tiles width and height instead of a set WxH? I dont know how well this will work, but were the blocks that will have different values will be blocks the player/NPCs can’t step on then it may not be an issue, again I’m not even sure if that is a viable solution.  My code in case you want to see how I'm currently handling it:  (pastebin) (edit:im not handling it, meant how I'm currently doing the map)  tl;dr - how can I take take my 2d looking tile map and change it to have different size tiles? or is there a better way to do what I am trying to do? Is there a name of a method that I should be looking for while googling? I imported the coordinates from an Excel table. They show as single points as intended. Now I want to add columns to the attribute table, but I'm not able to edit it. I wanted to import the following columns from Excel as well, but that doesn't seem to work either.  Sorry for the noob-question, but I couldn't find anything relating the theme through google and such.  Ok, with pictures then: I want to use that to add columns with information to these       You see, the small pensil thingy is greyed out, so I can't edit anything in this layer and I can't add any information to my points.  If I try Ale suggestion I get this error: What's the process to change a site logo? Who is responsible for defining a site logo? I mean, there are moderators of the site that define it, or stackexchange defines it? An explanation of Hawking Radiation Could someone please provide an explanation for the origin of ? (Ideally someone who I have been speaking with on the h-bar)  Any advanced maths beyond basic calculus will most probably leave me at a loss, though I do not mind a challenge! Please assume little prior knowledge, as over the past few days I have discovered that much of my understanding surrounding the process as virtual particle pairs is completely wrong. How to add an extra level of sections with headings below \subsubsection Last night I was updating my Ubuntu 10.04 to either 11.04 or 12.04 (I'm not entirely sure which one it was, but I think it was 11.04) and I left it overnight to finish. When I came down in the morning, I noticed that the computer hadn't barely begun installing the updates! The update thing said that my computer was on the installation step, but it was still preparing the libc bin (or whatever its called) . It's like the first part of the step installing the upgrades. So, I stupidly restarted the computer, and when it rebooted, I got five options on a purple screen:  Ubuntu, with Linux 3.0.0-28-generic Ubuntu, with Linux 3.0.0-28-generic (recovery mode) Previous Linux versions Memory test (memtest86+) Memory test (memtest86+, serial console 115200)   Long story short, not matter which way I enter Ubuntu to try and reinstall the updates using the root thing (it's like terminal) after I enter sudo apt-get update it basically said it failed to fetch every url link, and it then says:  W: Some index files have failed to download. They have been ignored, or old ones used instead. W: Not using locking for read only lock file /var/lib/dpkg/lock E: dpkg was interrupted, you must manually run 'sudo dpkg --configure -a' to correct the problem.   So, I entered sudo dpkg --configure -a, and it says:  dpkg: error: unable to access dpkg status area: Read-only file system"   I'm left like wtf, and I really just some guidance here. I just got this laptop, and I can not afford a new one if I broke this one. How should self-deletes be evaluated in the question ban algorithm? A self-deleted post is often not a low-quality post and may simply be a mistake noticed by the user, unlike moderator- or community-deleted posts which may indicate serious problems that the user may not recognize or may have intentionally posted.  Should they carry less weight in the question ban algorithm? The   echo one; echo two &gt; &gt;(cat); echo three;    command gives unexpected output.  I read this:  and many other articles about process substitution on the internet, but don't understand why it behaves this way.  Expected output:  one two three   Real output:  prompt$ echo one; echo two &gt; &gt;(cat); echo three; one three prompt$ two   Also, this two commands should be equivalent from my point of view, but they don't:  ##### first command - the pipe is used. prompt$ seq 1 5 | cat 1 2 3 4 5 ##### second command - the process substitution and redirection are used. prompt$ seq 1 5 &gt; &gt;(cat) prompt$ 1 2 3 4 5   Why I think, they should be the same? Because, both connects the seq output to the cat input through the anonymous pipe - .  Question: Why it behaves this way? Where is my error? The comprehensive answer is desired (with explanation of how the bash does it under the hood). I'm using the following command to embed images in my LaTeX document (which is then rendered as PDF)  \newcommand{\img}[4]{   \begin{figure}[!hbt]     \centering       \vspace{1ex}       \includegraphics[width=#2]{images/#1}       \caption[#4]{\label{img.#1} #3}     \vspace{1ex}   \end{figure} }   When creating a href to an image, the viewer jumps to the image caption instead of the top of the image. How do I change my \img command so it still puts the caption below the image but links to it jump to the image itself?