To prove $\sup |x^n g(x)| $ tends to zero as x tends to infinity In this particular problem,I am unable to understand how to prove the claim of sup(x) part. How can we use the information that g(1)=0.I need something to start off...
Evaluating the supremum of a function as n tends to infinity Problem: Let $g:[0,1]\to\mathbb{R}$ be a continuos function such that $g(1)=0$. Show that $\sup\limits_{x\in[0,1]}|x^ng(x)|\to0$ as $n\to\infty$.  My approach: Let $f(x)=x^ng(x)$. Then, $f(x)$ is continuos on $[0,1]$. Using the extreme value theorem we can say, that $|f(x)|$ is bounded on $[0,1]$ and attains its bounds, i.e., $\exists x_0\in[0,1]$ such that $|f(x_0)|=\sup\limits_{x\in[0,1]}|f(x)|$.  Now, $|f(x_0)|=|x_0^ng(x_0)|$. If $x_0=0$, then we are done. So, let us assume $x_0\neq0$. By the Archimedean property of $\mathbb{R}$, $\exists$ a $y\in\mathbb{R}$ such that $y=x_0^n\leq1$, since $x_0\in[0,1]$.   Hence, $\lim\limits_{n\to\infty}\sup\limits_{x\in[0,1]}|x^ng(x)|=\lim\limits_{n\to\infty}\sup\limits_{x\in[0,1]}|f(x)|=\lim\limits_{n\to\infty}|f(x_0)|=\lim\limits_{n\to\infty}|yg(y^{\frac{1}{n}})|\leq\lim\limits_{n\to\infty}|g(y^{\frac{1}{n}})|=g(1)=0.$  I will be glad if anyone can tell me if I have done any mistake and if so please help me to correct it.  Thank you.
Understanding Bash's Read-a-File Command Substitution I am trying to understand how exactly Bash treats the following line:  $(&lt; "$FILE")   According to the Bash man page, this is equivalent to:  $(cat "$FILE")   and I can follow the line of reasoning for this second line. Bash performs variable expansion on $FILE, enters command substitution, passes the value of $FILE to cat, cat outputs the contents of $FILE to standard output, command substitution finishes by replacing the entire line with the standard output resulting from the command inside, and Bash attempts to execute it like a simple command.  However, for the first line I mentioned above, I understand it as: Bash performs variable substitution on $FILE, Bash opens $FILE for reading on standard input, somehow standard input is copied to standard output, command substitution finishes, and Bash attempts to execute the resulting standard output.  Can someone please explain to me how the contents of $FILE goes from stdin to stdout? Sometimes I need to look up certain words through all the manual pages. I am aware of apropos, but if I understand its manual right, it restricts search to the descriptions only.     Each manual page has a short description available within it. apropos searches the descriptions for instances of keyword.   For example, if I look up a word like 'viminfo', I get no results at all...  $ apropos viminfo viminfo: nothing appropriate.   ... although this word exists in a later section of the manual of Vim (which is installed on my system).      -i {viminfo}                When  using  the  viminfo file is enabled, this option sets the filename to use, instead of the default "~/.vim‐                info".  This can also be used to skip the use of the .viminfo file, by giving the name "NONE".    So how can I look up a word through every section of every manual? What are the differences between 32-bit and 64-bit, and which should I choose? What is the difference between 32-bit and 64-bit Ubuntu? I've heard the 64-bit platform performs better and can detect more than 4GB of RAM. Also, while some apps haven't ported to 64-bit yet, ia32-libs lets a 64-bit machine run them.  If so, why not promote 64-bit over 32-bit? How do I catalog files on several external hard drives that I want to store off-line? I have quite a few external hard drives that I want to store off-line, and possibly off-site.  The contents is not saved in a structured manner, however; basically  there's lots of project data for lots of projects.  Is there any software for Windows XP that will catalog all the hard drives, so I can search the catalog, find the files, and get the ID of the disk that has the content? "Subject, verb, direct object, object complement" versus "subject, verb, indirect object, direct object" Reading English Grammar (HarperCollins College Outline, published by HarperResource, an imprint of HarperCollinsPublishers) I found a chapter (Sentence Basics) that explains that in English there are different sentence patterns; in particular   Subject, verb, direct object, object complement Subject, verb, indirect object, direct object   The chapter reports the following example for the patterns:     Alice called Charles Professor Miller.   Alice gave Charles the rabbit.   What is the difference between the two kind of sentences? What is the difference between the object complement in the first case, and the indirect object in the second case?  PS: The question, which could appear silly, comes from the fact in my first language there is only the object complement. First quick glance at a dataset Please pardon my ignorance, but...  I keep finding myself in a situation, where I'm faced with a bunch of new data I managed to find. This data usually looks something like this:  Date     Number1  Number2  Category1  Category2 20120125      11      101        Dog      Brown 20120126      21       90        Cat      Black 20120126      31      134        Cat      Brown (...)   Usually at first glance I can't really tell if there are any trends here. The correlations between the various columns may not be very significant, but I would be delighted if I didn't have to manually create a plot for every possible combination of columns/categories.  Is there a tool out there which would accept a table of data along with information which columns should be treated as numbers, dates and categories and then proceed to plot:   correlations between each two numerical columns correlations between each two numerical columns, with separate trend lines for each category each number column as a time series, each number column as a time series, separated by category, etc.   In the end this would generate a large number of plots, most of which would show only noise. Ideally, the tool could score the plots by correlation and in the end display a slideshow starting with the highest scoring plots. This would be a very imperfect, but useful first glance at the dataset.  So? Is there a tool everyone uses for this and I just don't know about it, or is this something we need to make? What counts as "observation" in Schrödinger's Cat, and why are superpositions possible? So if I understood correctly, Schrödinger's Cat is a thought experiment that puts a cat inside a box, and there's a mechanism that kills the cat with 50% probability based on a quantum process. The argument is that the cat now must be in a superposition of dead and alive.  This makes sense at first, but the state of the cat inside the box will affect the outside world in an observable way, right? For example if the cat dies, it might meow loudly which would be audible. If it didn't meow, it would produce a thud on the ground when it dies. And even if the ground was very solid, the redistribution of mass inside the box will affects its gravity field which means the whole universe theoretically immediately observe's the cat's death.  So extending this argument to all superpositions, the different states would cause different effects on the rest of the universe, usually a slight change in the gravity field is the minimum. This gravity perturbation would propagate throughout the universe, and even all the experimenters go to sleep with thick, thick earplugs, somebody or something in the universe is going to inadvertently observe the event and the superposition immediately collapses. Thus superpositions cannot exist beyond an extremely short amount of time.  What's wrong with my reasoning? I need to use the setting Disable tiled raster layer exports from the GUI in my Python code, but I can't seem to find it.  exporter = QgsLayoutExporter(layout)     exporter.exportToPdf(path, QgsLayoutExporter.PdfExportSettings())   How can I do this? How to make read-only file system writable? At some point, the filesystem on my digital audio player has become read-only. I cannot copy files into it or remove files on it.  Are there some possible reasons for the player's file system to change permissions in this way?  I tried using chmod:  $ sudo chmod a+rw SGTL\ MSCN/  chmod: changing permissions of `SGTL MSCN/': Read-only file system   SGTL MSCN is the mount point of the digital audio player.  What else can I try to make it writeable? LaTeX code sniffer and/or beautifier? I'm looking for a LaTeX code sniffer and/or code beautifier, because I'm creating a CI server for my LaTeX books and I wanted a few things. :D function main() {    Hello(); }  function Hello() {   // How do you find out the caller function is 'main'? }   Is there a way to find out the call stack? My MacBook Pro has a flashing question mark in a folder against a grey background My MacBook Pro shows a grey screen with a flashing question mark in a folder. I tried connecting it to an older MacBook using firewire cable. I restarted the Macbook Pro in Target Mode. Now I see Thunderbolt and Firewire icons flashing and moving in a grey background. There is a green power icon at the bottom as well. I am not an expert even though I have been using Macs for ages.  How can I fix this issue so I can recover/backup things in the MacBook Pro hard drive? It  to disable internet search results in start menu searches before the Windows 10 Creators Update, but I've just clean installed a new Windows 10 Pro image and that setting is gone. Is there some way to disable it through editing the registry/etc? What's the standard "roadmap" to learning quantum physics? I'm really interested in quantum physics and would like to learn more. However, I don't know where to start and in what order I should learn things. So, ideally I'm looking for some sort of roadmap of what to learn. What physics topics do I need to know to start learning about quantum mechanics? (In addition to the mathematical topics mentioned at )  My current knowledge is mostly popular science stuff, like tv shows on Discovery Science and National Geographic Channel. So I have a basic understanding of some basic principals. There's also a recent lecture from Brian Cox that I have watched which gave a bit more in-depth information. Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$ Plotting solutions to NDSolve What issues / pitfalls must be considered when overriding equals and hashCode? I have a many-to many relationship between three objects. LGFProducts, PartsinProducts, and Parts. PartsinProducts is the junction object. I want to be able to get data like the names of the parts that are related to the selected LGFProduct.   I understand that parent-child relationship can only traverse one level down while grandchild-grandpa can travel 5 levels at most. What is the parent-child or child-parent in a many-to-many relationship?  I assumed that I should be traversing upwards from Parts to the junction object partsinproduct and then LGFProducts? But my friend who is not a Salesforce expert but has knowledge in CS said that I should do a parent-child query (of LGFProducts to partsinproduct) to child-parent relationship query (PartsinProduct to Parts). Or perhaps both of these options are wrong and another is appropriate.   Also, I am having a little trouble traversing for custom objects. I understand that I need to append the letter r instead of c but get confused. For example, this is my attempt at a three level upward traversal that fails.   SELECT Id, Name, Parts__r.PartsinProducts__r,LGFProducts__r.Name FROM Parts__c   Also if three levels of upward traversal are needed, could one provide an example with custom relationships? As seen above, I am having difficulty successfully querying a three level upward traversal statement for custom objects.  This is my attempt at the parent-child relationship query of LGFProduct to PartsinProducts:  SELECT Name,   (     SELECT Name     FROM PartsinProduct__r   ) FROM LGFProduct__c    I am then given this error: Why is my variable local in one 'while read' loop, but not in another seemingly similar loop? My wife was just issued a multiple entry visa by France, which is good for one year.  Our first trip will be to France, the issuing country.  My question is: after we leave the EU and return, must future trips also start in France? Or may we go directly to (say) Greece from the US on a later visit?