Prove that $\limsup(a+b) \leq \limsup(a) + \limsup(b)$ and $\liminf(a+b) \geq \liminf(a) + \liminf(b)$ Prove that $$\limsup(a+b) \leq \limsup(a) + \limsup(b)$$ and $$\liminf(a+b) \geq \liminf(a) + \liminf(b)$$ where $a,b:\mathbb{N}\to \mathbb{R}$ are two bounded sequences.  I'm not sure where to begin on this proof.
Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$ I am stuck with the following problem.     Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$   I was thinking of using the triangle inequality saying $$|a_n + b_n| \le |a_n| + |b_n|$$ but the problem is not about absolute values of the sequence.  Intuitively it's clear that this is true because $a_n$ and $b_n$ can "reduce each others magnitude" if they have opposite signs, but I cannot express that algebraically...  Can someone help me out ?
How can I convert my Java program to an .exe file? Are any of the alien languages in Star Wars real Earth languages? Are any of the alien languages (other than English and binary) in Star Wars movies or TV shows real Earth languages? how can i show site logo, name and slogan Separately in custom area in my site theme, I now use this code to display theme, buts this code don't allow me to edit each of them Separately.  {{ page.header }} How to implement draggable tab using Java Swing? How do I implement a draggable tab using Java Swing? Instead of the static JTabbedPane I would like to drag-and-drop a tab to different position to rearrange the tabs.  EDIT: . I'm trying to mount my HDD,  /dev/sdb, which is connected via an enclosure but every time I try do so I get an error message   mount: /dev/sdb: can't read superblock   How do I solve this problem and mount my HDD? Mod shows up with OP background when viewing deleted comments Why are there no Pokémon near me? How did the transdimensional Edo god (Edo lord) come to be the overseer of the Edo? If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p. If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.  I suppose this would not be true if p $\equiv$ 1 (modulo 4).  To prove something is a non-square I find to be tricky. It's difficult to see any straightforward way to do this using only the definition of congruence for example. Limit at infinity of $x^{1/x}$ I encountered a problem: calculate $\lim\limits_{x\to\infty} x^{1/x}$  My solution is quite weird and I find no document explaining whether I am right or wrong!  My solution:  As $\lim\limits_{x\to\infty} 1/x = 0$  then $\lim\limits_{x\to\infty} x^{1/x} = \lim\limits_{x\to\infty} x^0  = 1$  My question is: Whether I am right or wrong, and could you please explain why? Extracting Z data from point layer using ArcGIS Desktop? I have created a point layer with ~150 points. I have a DEM beneath it, which is where I have taken the X and Y values from.   Does anybody know how I can retrieve the Z values, without using the info button?  I am using ArcGIS Desktop 10. I have licences for the spatial analyst etc tools. SEVERE: A child container failed during start java.util.concurrent.ExecutionException: org.apache.catalina.LifecycleException: Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory? Combining the entries of two lists SHM with acceleration at mean position Suppose we have an equation of  motion as $$\frac{d^2x}{dt^2} = -kx + c,$$ then can it be called a SHM? Since acceleration is still proportional to displacement. But then, how will we define the mean position? $e^{mx}$ in solving second order differential equations In a book I am reading on differential equations, the author writes about the solution to a homogenous, linear, second order differential equation with constant coefficients. The author says something like, "Let us suppose that the solution is of the form $y=e^{mx}$ " . After this the author introduces the characteristic equation of a differential equation of the form mentioned above, and proceeds to describe how to solve it w/ undetermined coefficients, variation of parameters, etc.   How did mathematicians first come up with this "assumption" that the solution was of the form $y=e^{mx}$? And, are there any other forms of solutions for these types of equations? Force fsck.ext4 on reboot, but really "forceful" One of my Ubuntu 10.04 servers is giving me trouble. When I run fsck.ext4 -n /dev/sda5 it tells me there are errors in the free inode count, free block count, and more.  I have tried:  touch /forcefsck   Also tried:  shutdown -rF now   and still, after reboot, I see errors.  I also just checked on my eeePC netbook, Ubuntu 10.10, and have the same issue!  How can I force a really "forced" "forceful" "seriously fix my filesystem" fsck of the "/" filesystem on reboot?  Clarification: I run fsck.ext4 -n because it's a mounted filesystem, to check if there are errors. This tells me that there are. I thought that the automatic fsck every 30 mounts during the boot-up process is precisely to take care of errors in the root filesystem. But it doesn't do it in my case. I could reboot with a LiveCD and fix the errors, and then reboot again, but that's some serious downtime for a live server. A reboot, auto fsck, then continue booting is much more sustainable on a live server, and I believe should be the right behaviour.  Additional info: Here is the output. It looks like something that the autofsck would fix, doesn't it?  root@server:~# fsck.ext4 -n /dev/sda5 e2fsck 1.41.11 (14-Mar-2010) Warning!  /dev/sda5 is mounted. Warning: skipping journal recovery because doing a read-only filesystem check. /dev/sda5 contains a file system with errors, check forced. Pass 1: Checking inodes, blocks, and sizes Pass 2: Checking directory structure Pass 3: Checking directory connectivity Pass 4: Checking reference counts Pass 5: Checking group summary information Free blocks count wrong (1849368, counted=1948909). Fix? no  Free inodes count wrong (545504, counted=552134). Fix? no   /dev/sda5: ********** WARNING: Filesystem still has errors **********  /dev/sda5: 116752/662256 files (0.2% non-contiguous), 795324/2644692 blocks What does "if not" mean in the given sentence Let there be given this sentence (which came from an ):     The contest has become personalised, if not bitter.   Then what does the phrase if not mean?   Seeking after is a general guide or rule of such usage. Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$ Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$.  The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and the groups may ask other people for hints. This is how I got involved, but of course asking on MSE would have been cheating)  I found three solutions: I am explaining below what I did. I would like however to know if there is a more general approach. Is there a way to make comments in the node editor? I'd like to have a node like the Frame node but one that can have lots of text, not just a header. I've looked and it looks like there isn't any such capability so far, so I was wondering how easy it would be for someone to create a custom Comment node as an add-on or script (I don't know the differences between the two). I myself would love to leave detailed comments in complex node trees so that I can figure out what I was doing/what to do when I come back to them months later.