If ${a_n}$ converges, then ${a_{2n}}$ converges proof question
Prove: If a sequence converges, then every subsequence converges to the same limit.
How to prove that $4^m + 5^m$ is divisible by 9 when m is an odd number I am trying to use congruence theorems, specifically Euler's Theorem for a proof. How to count the number of occurrences of an element in a List I have an ArrayList, a Collection class of Java, as follows:  ArrayList&lt;String&gt; animals = new ArrayList&lt;String&gt;(); animals.add("bat"); animals.add("owl"); animals.add("bat"); animals.add("bat");   As you can see, the animals ArrayList consists of 3 bat elements and one owl element. I was wondering if there is any API in the Collection framework that returns the number of bat occurrences or if there is another way to determine the number of occurrences.  I found that Google's Collection Multiset does have an API that returns the total number of occurrences of an element. But that is compatible only with JDK 1.5. Our product is currently in JDK 1.6, so I cannot use it. How do I remove duplicate "Open With" context menu items in Finder.app? For whatever reason, the "Open With" context menu in Finder is always listing every app four times (exactly).     I've read about (and tried) rebuilding the Launch Services database to restore the context menu to its initial state, but nothing seems to work. I've also used Onyx.app to clear user and system caches, but I'm still left with four repetitions of each app under the "Open With" service. I have restarted the system a few times just to be sure it was not some temporary corruption of the list.  I'm running OS X 10.6.6 on a 2011 17" MacBook Pro with a fresh install of OS X (i.e. I didn't use Migration Assistant.app or a Time Machine backup). I did, however, sync all my preferences and other files using MobileMe like I always do, so I imagine it's feasible that a preference file somewhere is causing problems? I didn't notice whether this was happening before or after my MobileMe sync.  How can I fix this so one app shows? I'm trying to make a page where a user provides an ID of a attachment and the site will set the headers and deliver that file instead of a 404 page. so /download/1754 would lookup the attachment with that Id and output it for viewing/download depending on the file type.  I just can't think of a wordpress friendly way to do this from inside a plugin. I considered using add_rewrite_rule but couldn't see a good way around it, also considered checking $_SERVER['REQUEST_URI'] to deliver the content before any HTML but then I'd have to stop the page from rendering a 404 and I was not sure of a Wordpress friendly way to do this.  My current idea was to have my plugin make a download page, assign a template and have that template run the logic instead of my plugin class. while I have used this before for another situation it was very messy as I had to write a lot of code to force users to be unable to update or edit the file and recreate/delete it on activation.deactivation. so ideally I'd like to avoid that. I have just updated Windows 7 to Windows 10.  My touchpad scolling (right-side) is not working anymore.    I have no idea how to fix this in Windows 10.  I have hit the start button and typed "synaptic", but have found nothing useful.  I used to have an icon on the taskbar that showed where I was touching the touchpad.  I used to be able to right-click that icon to change settings, but Windows 10 has not brought that icon over.  Also, the asus website has no synaptic touchpad drivers available for Windows 10.  I have tried the enaltech drivers, but upon the installation atempt I am told that my hardware is not correct. Arduino-controlled slot cars - alternative to MOSFETs for speed control? I'm attempting to control the speed of two slot cars in a set using Arduino.    Each car is controlled by a controller with a variable resistor that looks like this:    They're wired in parallel to the same 17V DC power supply:    Simplified:    For Arduino control, I experimented with manipulating a single car's speed using a MOSFET. It worked really well. Using a PWM output on the Arduino, I was able to pulse the MOSFET on and off to throttle the current and change the car's speed. The behavior of the MOSFET seemed to be exactly the same as the variable resistor controller; the resistance, voltage, and current fluctuated the same way on various parts of the circuit.  Enter car B. I added a MOSFET to the setup and mimicked the MOSFET wiring for car A, grounding them both to the Arduino ground. The result looks like this:    Here's a sketch:    Close up, MOSFET A:    MOSFET B:    At first, all seemed well. I sent signals to MOSFET A and successfully changed Car A's speed. I stopped Car A and pulsed MOSFET B, and I was able to control Car B's speed just fine.  Then, I turned Car B to a constant speed, and sent a signal to MOSFET A while Car B was still running. Car A started, but Car B's speed immediately dropped as a result of running Car A.  I hooked a multimeter up across the rails of Car B and watched its voltage while it ran on its own. Then, sure enough, when Car A started running alongside it, Car B's voltage dropped dramatically.  I figured this was because of my wiring configuration, so I left the multimeter on Car B, and made one change. I replaced MOSFET A with the original slot car controller:    I started running Car B again using its MOSFET. The multimeter showed a steady voltage, so I began to squeeze Car A's controller (the variable resistor), and Car A's speed increased. This time, however, unlike using a MOSFET to control Car A, the variable resistor did not affect Car B's voltage at all. Car B remained at a steady speed the whole time.  Any idea why this happens? The MOSFET seemed to function the same as the variable resistor (empirically) with a single car, but had totally different behavior with two cars in parallel. Is it because the MOSFET is not really changing resistance to current, but just switching it on and off, so it opens up another path for current to flow fully every time it closes? Is there an Arduino-controllable alternative to a MOSFET that would fluctuate actual resistance like the variable resistor? Digital potentiometer? Servo hooked up to a potentiometer? ;) I'm really curious what the difference is and what causes the major differences in behavior. UnboundLocalError on local variable when reassigned after first use Prove the following in $S_n$:  Let $\alpha = (a_1,a_2,\ldots,a_s)$ be a cycle and let $\pi$ be a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),\ldots,\pi(a_s)).$     I'm not sure where to even start. How to override 'DefWndProc' in WPF Suppose we have $x^2+4x+3=0$  We differentiate both sides to get $2x+4=0$.  So $x=-2$  But $(-2)^2+4(-2)+3=0$   $4-8+3=-1$ is not equal to $0$.  Which tells that we cannot differentiate equalities. But in many proofs, like in integral calculus, differentiating both sides occurs. So, where is the fallacy? Join matrix based on conditions Best Practice for Forcing Garbage Collection in C# Rename files in directory according to name of directory I am trying to rename a bunch of files according to the names of their directory in Linux, SSH.  I have a directory called XYZ01smith. In it are four files called smith_5*.I need to add prefix XYZ01 to these files. But, the next directory is called XYZ02perry. In it are four files called perry_3*. I need to add the prefix XYZ02 to these files.  I have a loop to do this per directory:  for i in smith_5*;     do mv ${i} XYZ01${i}; done    But I have 50 directories (XYZ01name to XYZ50name), so I would rather be able to do this with one script for all 50. I am sure this should be possible, but I do not know how. It has been discussed before, but in light of some of the , I can't help but wonder what is accomplished by displaying the author of a given question or answer.  There is a , but if anything, I don't think it goes far enough.  If questions and answers are supposed to stand on their own merit, why do we need to show the author at all?  I'm not proposing any change to the way rep works; rep should still be accrued and abilities gained.  The only difference is that questions and answers would appear anonymous.  Does this better reflect the goals of SO?  If so, how far do you think the anonymity should be taken?  If not, what shortcomings am I missing? Understand PAM and NSS In the last days I have set up some Linux system with LDAP authentication and everything works fine, but there's still something I can't really understand regarding NSS and PAM, also after a lot of research. Citing:  NSS allows administrators to specify a list of sources where authentication files, host names and other information will be stored and searched for  and  PAM is a set of libraries that provide a configurable authentication platform for applications and the underlying operating system  What I don't understand is how PAM and NSS work and interact together. In  the architecture is explained pretty well: I configure PAM to use pam_ldap for LDAP accounts and pam_unix for local accounts, then I configure nsswitch.conf to fetch information from local files and LDAP. If I have understood correctly LDAP is used twice: first by pam_ldap and then by NSS which is itself called from pam_unix. Is that right? Is LDAP really used twice? But why do I need to configure both NSS and PAM? My explanation is that PAM performs different tasks than NSS and it is used by other programs. But, then, it should be possible to use only NSS or only PAM, as I have read in . So I experimented a bit and I have first tried to delete LDAP from the nsswitch.conf (and the authentication stopped to work as if only pam_ldap is not enough to do the job). Then I re-enabled LDAP in NSS and I deleted it from the PAM configuration (this time everything worked fine, as if pam_ldap is useless and NSS is enough to authenticate a user). Is there anyone who can help me to clarify this? Many thanks in advance. UPDATE I've just tried something now. I removed again all the pam_ldap entries in all pam configuration fields and I have also removed shadow: ldap from nsswitch.conf. As now in all the system there are only the lines: passwd: ldap files and group: ldap files in nsswitch.conf. Well... the login with LDAP users works perfectly, those two lines (plus /etc/ldap.conf) are enough to configure LDAP auth. From my knowledge PAM in independent from NSS, but my tests showed it's not. So I ask myself is it possible to completely disable NSS and use only PAM? This is a question from the book Methods of Real Analysis by R. R. Goldberg.     If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$.   I don't have any idea how to start working on this problem. Please help. Thanks. How to solve differential equations of the form $f'(x) = f(x + a)$ What could one do to find analytic solutions for $f'(x) = f(x + a)$ for various values of $a$?  I know that $c_1\sin(x + c_2)$ is solution when $a = \frac{1}{2}\pi$, and of course $c_1e^x$ when $a = 0$.  For instance, is there a function satisfying $f'(x) = f(x + 1)$?  What about negative or imaginary $a$?  Is there possibly a generalization? Why does Jack let Walt go? In Breaking Bad S5 E14, Jack not only lets Walt go, but also leaves a barrel of money to him after killing Hank. I don't understand why he would do this. Jack says that, with 7 barrels of money, he doesn't need to cook meth any more, so there is no reason to keep Walt (the witness of the killing) alive, let alone to give him so much money. What's the rationale behind this? Or it's just mercy or respect? How to change the shadow color in Blender Cycles? I want to change the color of the shadow casting by a cube to green. While it's very easy to do it in Blender Internal (you select the lamp, go to the Lamp data header and change the shadow's color in the Shadow panel) in Cycles there're no such options. How to achieve it in Cycles? Why teach linear algebra before abstract algebra? Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra?  I'm asking this because personally it seems to be much easier to understand the architecture behind linear algebra as supposed to simply solving problems after a course in abstract algebra.