$\{{x_n}\}_{n \geq 1}$ be a sequence such that $|x_{n+1} - x_n| \leq \frac{1}{2} |x_n - x_{n-1}|.$ Prove ${x_n}$ is convergent
$\left|x_{n+1} - x_n\right| < C\left|x_n - x_{n-1}\right|$ prove this is Cauchy
Schengen Visa Refusal: Justification for the purpose and conditions of the intended stay was not provided Standardize Training and Validation Data How / can you connect Ethernet devices together without magnetics? How do you wire it up? What sorts of practical limitations are there on doing this?  To clarify, I'm not trying to connect two computers here. I'm thinking specifically of creating a small network of microcontrollers on a single board. Lets say the total length of a network segment is no longer than 6 inches. This would be for 10BaseT or at most 100BaseT. How many objects are being created? Identification card for foreigners in trains in Germany I want to go to Berlin to Gottingen by train. I found that to avail myself of the &ldquo;savings fare&rdquo;, I have to book a ticket at least one day in advance. And I saw a requirement on the Bahn website to have a identification card for online tickets.  I am a foreigner in Germany, so does my passport count as valid identification in German trains? How to change the chapter number to display as text (One, Two, ...)? When I want to register an account with Gmail now, it says "Google will send a text message containing a verification code to your mobile phone."  I don't want to give Google my mobile phone number!!  How do I get around this? Does it make sense to say, "The speed of light varies?"  Some may say right off the bat "Yes, it changes as a wave passes through a different medium."  However, I'd like to say no, because when I hear someone say the speed of light, I always think of the constant $c$ (unless the medium is specified to not be a vacuum, but then it isn't $c$ anymore), not the speed of a particular wave.  To me, it makes more sense to say something like, "The speed of a particular wave varies." What is the correct way to state this in the professional world? And in general, when professional physicists say "the speed of light," are they referring to the constant or the actual speed of the wave? There are many cases in maths where doing the same thing over and over again will bring you closer to a value. Derivation of polynomials is the striking example: no matter what you start with, you end up with 0. In a series expansion, the longer you continue the sequence the closer you get to the real value.  This concept applies in areas other than math (Wikipedia's everything-leads-to-philosophy phenomenon, for example). It's a little unrelated but it just proves the concept.  Hashing algorithms are just special math. Following logically, hash entropy supposedly decreases the more you hash a value. So is there a point at which any input value will give the same hash as any other?  (Let me just specify that we're not re-injecting any new data every time we hash, as if we're trying to increase the security of the hash by salting and re-hashing. It's strictly a matter of md5(md5(md5(... if that's the algorithm we're using.  I can see some problems with this already. Algorithms like md5 are designed to change their output radically when the input changes only by one bit of information. So even if a common hash for two values appears along the way, it itself will start changing. But then, does the fact that the output appears to be very different actually mean the possibilities are still many? Why is the well ordering principle counter-intuitive? I read  that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why this is so. According to Wikipedia, 'the well-ordering principle states that every non-empty set of positive integers contains a least element'. This is quite a straightforward definition. Is it the way you derive the equivalence of the Axiom of Choice and well ordering principle that makes it counter-intuitive or the definition of the least element? Ifstream read function doesn't load into vector My roommate put a hunk of beef roast in the slow-cooker for a good 6 hours.  It's a fairly large sized slow-cooker, at least 8 quarts.  The meat was cooked through but very chewy, and I never remember my mom having that issue with a slow-cooker.  My roommate tends to put only enough water to get halfway up the side of the roast, and always uses a slow-cooker 'bag' (another thing my mom never did).  What advice can I offer them to get a better outcome? Not an essential singularity The following is an old qualifying exam that I cannot solve:  Suppose $f$ is holomorphic on the punctured unit disk and $ \frac{f '}{f} $ has a simple pole at $0$.  Show that $f$ does not have an essential singularity at $0$. Watts P.M.P.O. and audio equipment power I am a beginner in electrical engineering, and to begin learning I decided to look into audio amplifiers. I have seen that the power of audio amplifiers is often described in the units of watts, watts RMS, and watts PMPO. So far, the only one of those units that I think I understand is watts. For instance, I just recently made . If I think: &quot;What is the power of my amplifier supposed to be?&quot;, then my thought process goes like this:  The transistors I used are the TIP120, which have a maximum rated power of 65W. Then, I decide that 60W is a safer maximum which would be reached if my amplifier runs 19V through one of my 6Ω speakers. Therefore, the power of my amplifier is 2x60W.  Then, I look at the datasheets of audio amplifier ICs, like the TPA3116 that says: &quot;50W into a 4Ω load at 21V&quot;, which makes me conclude that in that setting it'll output 14V maximum into the speaker, which I think makes sense. But then, I look at the datasheet of the TDA7294, and they talk about &quot;RMS Power&quot; and &quot;RMS Music Power&quot;, units which I couldn't find clear information about. Then, I look at home audio systems, like a Panasonic Stereo System I have which has &quot;9500 Watts P.M.P.O.&quot; written on the front. 9500W? That sounds stupidly high to me, but then I look again and it has 300W written on the back of it. If I'm correct, maybe a 1kW home stereo system could make sense, but a 2kW or greater one would not exist. So... Are my conclusions about  and the TPA3116 correct? Are my conclusions about audio equipment power correct? Are WattsPMPO, WattsRMS and RMSmusicpower, real useful units? Why not introduce a donate button in Stack Exchange sites? When using sudo powers and calling gedit the top level menu with File Edit View Search Tools Documents Help is missing.  If I create user ID root and sign in once and call gedit will it create the necessary user configuration files that might fix this?  Will this also make the annoying error messages go away whenever I use sudo, gksu or pkexec as a regular user with elevated privileges togedit ?  Will there be other benefits with Nautilus and other Gnome derivative Ubuntu applications?  NOTE: by sudo powers you can google pkexec gedit 3.5k hits, gksu gedit 40k hits or sudo gedit 500k hits. I'm in the minority using the first method but I believe it will become the standard in Ubuntu 17.04.    June 12, 2017 Update Here are a list of errors you get when using pkexec gedit:  (gedit:13003): Gtk-WARNING **: Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.gnome.SessionManager was not provided by any .service files  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-spell-enabled not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-encoding not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-position not supported   On this  there is a commitment to fix these distracting messages as of June 5, 2017. No indication is given for when the fix will be upstream nor if the functionality will be implemented or the error messages will simply disappear.  Note the bug report was filed over a year ago and over that time it only effects 18 people. Alien invasion movie, giant robot aliens with long legs capturing people So I saw this movie about 10 years ago, when I was about 5 years old. Since I was really young, my memory of it isn't really that great.   I think it may have been an alien invasion movie, but the only scene I remember is that there were these giant robot-alien things:   I think they had several long legs. They had a big glass dome on top of them They would grab civilians and drop them in their glass dome.   That's all I remember. If you could please help me out, that would be great.   Again, I was about 5 years old when I watched it, so my memory of it isn't that great, but I hope you know what I'm talking about. Thanks so much :) I'm confused. Running Fedora Linux, lscpu yields:  Architecture:            i686 CPU op-mode(s):          32-bit, 64-bit ...   But when I try to install a 64-bit program (Chrome) I get error like:     Package /....x86_64.rpm has incompatible architecture x86_64. Valid architectures are ['i686', 'i586', 'i486', i386']   I'm less interested in being able to install Chrome and more interested in why lscpu says that my CPU can run in 64-bit mode; clearly this can't mean I can run 64-bit programs. Can anyone clarify? The most efficient way to implement an integer based power function pow(int, int) What is the most efficient way given to raise an integer to the power of another integer in C?  // 2^3 pow(2,3) == 8  // 5^5 pow(5,5) == 3125 Finding limit of a sequence in product form \begin{equation} \prod_{n=2}^{\infty} \left (1-\frac{2}{n(n+1)}  \right )^2 \end{equation}  I need to find limit for the following product..answer is $\frac{1}{9}$.  I have tried cancelling out but can't figure out.  Its a monotonically decreasing sequence so will converge to its infimum.. how to find the infimum?