Let be $f:[0,\infty[ \rightarrow \mathbb{R}$ a decreasing and continuos, $f(x)&gt;0 , \forall x$. If $\displaystyle \int_0^{\infty}f(x)dx$ converges then $\displaystyle \lim_{x \rightarrow \infty} xf(x) = 0.$  How to prove that?  Ok, I know that if $G(x) = \displaystyle \int_0^{x}f(t)dt $ then $ \displaystyle \lim_{x \rightarrow \infty} G(x) = L$, How can I use that for arrive in $xf(x)$??
Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be a monotone decreasing function defined on the positive real numbers with $$\int_0^\infty f(x)dx &lt;\infty.$$ Show that $$\lim_{x\to\infty} xf(x)=0.$$   This is my proof: Suppose not. Then there is $\varepsilon$  such that for any $M&gt;0$ there exists $x\geq M$ such that $xf(x)\geq \varepsilon$. So we can construct a sequence $(x_n)$ such that $x_n \to \infty $ and $x_n f(x_n ) \geq \varepsilon$. So $$\frac{\varepsilon}{x_n}\leq f(x_n) \implies \sum_{n\in\mathbb{N}}\frac{\varepsilon}{x_n} \leq  \sum_{n\in\mathbb{N}} f(x_n) \leq \int_0^1 f(x)dx.$$ So we get a contradiction.  I feal like I have the correct idea but some details are wrong.  Any help would be appreciated.
LaTeX seems to be having problems handling nested optional arguments. I've included a minimal code example that generates the error:  \documentclass[12pt,oneside]{book} \begin{document}  \newcommand{\termbad}[2][]{#2#1} \newcommand{\termgood}[2][]{#2#1}  \newcommand{\up}[1]{#1} \newcommand{\dd}[1][]{#1}  \termgood{\up{Tart}}{Pop} \termbad[\up{Tart}]{Pop}      \termgood{\dd[Tart]}{Pop} \termbad[{\dd[Tart]}]{Pop} %\termbad[\dd[Tart]]{Pop}   \end{document}   If you run this code, it works. If you remove the % in the second to last line, it fails with:  Runaway argument?  Tart{Pop} \par \par \end {document}  !File ended while scanning use of \dd.   After further tinkering, I have discovered the problem: LaTeX is reading the ] for the inner optional argument as the end of the outer optional argument and then passing a malformed argument in. So it is passing in \dd[Tart as the optional argument to \termbad instead of \dd[Tart].  This still feels like a bug to me. Nesting braces in arguments works, brackets should as well. The following are workarounds:   Enclose the inner optional argument in braces (as in the line before %). Redefine your inner function to require its argument instead of using optional arguments. Redefine your outer function to require its argument instead of using optional arguments.   Any one of the three works around the issue.  (Reposted here; question was originally posted to stackexchange and then closed instead of being migrated.) Dragging layer into new file changes its size Let's say I have a file that's 4'x12' and represents a space on an actual wall. I want to open certain files (of different sizes) and drag each file into the large file in order to arrange the images correctly on the computer before physically hanging the framed finals.  When I drag a new image into the larger file, the new image's size and the size of its copy don't match. That is, Say I drag an 8x10 into the large file, well on the large file (trying to control size with guides and rulers) the new image is some other size, but not the right one.   What am I doing wrong? How do I make sure that when I click and drag the new image that it will be properly represented? fitch style proof problem regarding rule of inferences Okay I have the following fitch proof almost figured out, from here I know I have to move forth in conlcuding but I don't know what I am missing  (2 premises)  S → (R ∨ P)  P → (¬R → Q)   my work below  new scope assume S   Rule: > elim  R ∨ P  New scope assume R  Q∨R     Rule: V intro  End scope – (new scope assume P)  ¬R → Q         Rule: > Elim  my confusion comes where I go from here  Q∨R            rule Taut Con ??  Goal:├ S → (Q ∨ R) Without using any non-standard (Windows included) utilities, is it possible to download using the Windows command line?  The preferred version is Windows XP, but it's also interesting to know for newer versions.  To further clarify my question:   It has to be using HTTP The file needs to be saved Standard clean Windows install, no extra tools   So basically, since everybody is screaming , I want simple Wget functionality, without using Wget. Is an irrational number times a rational number always irrational?  If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number? I have been reading about the new Salesforce async triggers. Currently, I tend to handle these sort of needs with a trigger that passes data to a 'Handler class' that would evaluate if a certain change has occurred. If it has then we pass those Ids to a queueable class for further processing.   What advantage do CDC and async triggers offer over using queueable or batch apex? When testing the following code:  from osgeo import gdal import json   def read_file(filename):     f = gdal.VSIFOpenL(filename, "rb")     if f is None:         return None     content = gdal.VSIFReadL(1, 10000, f).decode('UTF-8')     gdal.VSIFCloseL(f)     return content   gdal.VectorTranslate('out.json', """{   "type": "FeatureCollection",   "features": [       { "type": "Feature", "geometry": { "type": "Polygon", "coordinates": "coordinates":[[[-188.56933593749997,61.58549218152362],[-173.9794921875,61.58549218152362],[-173.9794921875,66.00015035652659],[-188.56933593749997,66.00015035652659],[-188.56933593749997,61.58549218152362]]] } },   ] }""", format = 'GeoJSON', layerCreationOptions = ['RFC7946=YES', 'WRITE_BBOX=YES'])  got = read_file('/vsimem/out.json') gdal.Unlink('/vsimem/out.json')  print json.loads(got)   from this , the file out.json is empty. The folder is with all permissions for writing allowed. When testing the method gdal.OpenEx, is possible to load an object, but still the method gdal.VectorTranslate returns an empty geojson when trying to pass this new loaded object as a srcDS object. Any hint on solving this issue? The new ask page is now live on the network! Diamond moderator rollbacks can resurrect blacklisted tags How to use GCC 4.7 in place of 4.6? So I'm working on a project and I wanted to use some C11 features for C. To use those features GCC 4.7 is required, so I went to synaptic and looked up 4.7, and I found the packages gcc-4.7-base so I installed it[1].   So now that gcc-4.7-base is installed I expected to be able to use 4.7, but no, when I do gcc --version it still shows 4.6.3 even more, when I try to directly call 4.7(via /usr/bin/gcc-4.7) it doesn't exist, though a gcc-4.6 executable does exist.  How do I "activate" GCC 4.7? I read , you do some crazy stuff with update-alternatives, but it doesn't work for me as the gcc-4.7 executable doesn't actually exist on my machine.   Notes: 1. I also saw gcc-4.6-base, so I figured I'd uninstall it, but synaptic attempted to remove every package I had installed and install a Java runtime, so for obvious reason I decided not to remove it. Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric. Probability of Various Combinations of Weather There's a 10% chance it will rain today. There's a 15% chance it will snow today.   What percentage chance is it that it will:   Both rain and snow (it can both rain and snow at the same time)? Neither rain nor snow? -   Raining and snowing are independent of each other.  EDIT: (ADDED)   Rain or snow but not both? Python mechanize javascript Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected?  This is not homework but is in response to working several problems where countable or uncountable many points are removed and the pattern always holds.  Thank you. Store table in separate filegroup in SQL Server I have big SQL Server database, with a lot of blob data in some tables.  For performance optimization, I want to move one table to separate hard drive.  I created new filegroup, and "row data" file.  But, I can't change any parameters in table properties: they are all disabled.  And I don't know a way to enable it.  Can anybody help me? Is there a reason, beyond historical reasons, that three phases has become the dominant number of phases?  I am aware of the advantages against one phase and two phase, namely the reduced amount of conductor needed, and that motors can provide torque when stalled (and less pulsation).  Is this solely due to diminishing returns, with only a small increase in smoothness of torque application, at the cost of increased complexity (increased number of wires (albeit of smaller CSA)).  To be clear, the phases are all evenly distributed, that is, five phases separated by 72 degrees. What is the difference between a gameplay programmer and a game designer/developer? Checkbox in listview control Can you have a multicolumn listview control where one of the columns is a checkbox?  Example code or links would be greatly appreciated.  I am using visual studio 2005 How can I convince new players to read and learn the basic rules before sitting down to play? Everyone in this story works mostly together. Some work friends and my wife were talking about playing D&amp;D, and I was appointed the DM. I've played the game once 20 years ago, so I've spent the past few months buying the resources and learning as much as I can about the game so I can make it as enjoyable for then as possible, we are yet to have our first session.  But to make it easier so I don't have to teach them literally everything I asked them to read the Basic Rules so they have some idea of what they are in for and the expectations of them and me are.  But they came back with "its too hard to read it, I learn by doing, that's what we have you for".  I'm super pissed by this, I've spent so much time and effort learning this game and they can't be bothered to put any effort in.  My question is should I let them know how much it's annoyed me, and reiterate they need to know stuff, or just ignore it and carry on? Are Lorentz transformations linear transformations? My textbook says that Lorentz transformations are linear transformations and present them as matrices. Lorentz transformations relate different coordinate systems with each other. It seems that coordinate systems are linear spaces, but coordinates are just labels for spacetime points, with no other structures attached. So what is the linearity all about?