So I think this is the last problem I have and I'm not thinking I'm doing it properly.     Let $a,b$ be real numbers and suppose for all $\varepsilon \gt 0, a \le b+\varepsilon$.  Show that $a \le b$.   Assume $b\lt a$.  Then $b+\varepsilon \lt a + \varepsilon$ and  $$a \le b+\varepsilon \lt a+\varepsilon$$ If I can show that there is a contradiction I can prove it, but I'm having trouble with the contradiction.
I am reading Tom Apostol's Analysis and come across this theorem.   Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon &gt;0$?   I don't doubt the proof in the book but I don't understand the intuition or geometric explanation behind this. Could somebody shed some light on this equation? I just started studying analysis on my own.$\ \ $
How should I detect unnecessary #include files in a large C++ project? There are 8 crew members, The leading member wants that only a crew of 5 people or more could open a safe he bought, To each member he gave equal amount of keys, and locked the safe with several locks.  What's the minimal number of locks he should put:   at least 5 crew members would be required to open the safe? Any team of 5 crew members could open the safe, but none of the 4 team crew members.   1,2 stands individually of course. I think the questions says it all! :)  By TeX, I mean any derivative of TeX as long as the code is clearly a derivative of TeX, i.e. that it could belong on TeX.SX (and not pure Lua for example).  I'll give two answers myself so you see what I mean. How do I run Command & Conquer Gold on Windows 7 64-bit (in XP mode)? I downloaded the first Command &amp; Conquer game  and burnt some CDs. I couldn't get the Setup.exe program to run in Windows 7 64-bit, as it must need some old DLLs. So I ran up the Windows XP Mode virtual machine, and was able to install it fine.   However, when I try to run the program, I just get an empty message box with an "OK" button. I've set the compatibility to Windows 95, and checked all of the options in the Compatibility tab, but it's not running. I also downloaded the patch suggested by EA, which is an updated version of the THIPX32.DLL file (copied into the C&amp;C installation folder).  Has anyone managed to get this working? write data line by line on iphone How to inject keystrokes via a shell script? I'm trying to inject keystrokes into the input daemon so as to simulate typing from a Bash script. Is this possible, and if so, how can I make it happen? MS Access to Oracle easy Conversion / Migration "Those who qualify will be awarded a certificate" or "those who qualify will be awarded certificates"? My interest in Blender comes from wanting to use it to model physical structures and produce architectural drawings; what I have so far is really impressive, but there doesn't seem to be too much information about my particular interest.   Are there any (Blender-) resources that would be particularly useful for architects? E.g. packages of standard 'elements' like windows, doors, roofs or whatever, or perhaps a discussion group specifically for architects? What books, journals, and electronic resources are most valuable for expanding knowledge of GIS? What books, journals, and/or electronic resources have you found most valuable for expanding your knowledge in the GIS field, and why? I have the following pieces of hardware I would like to network mostly for file sharing (media primarily): 1 gaming laptop, 1 i7 (2.3Tb storage) desktop, 1 P4 desktop (500g storage), and 1 250gig PS3.  I am looking to setup a linux host (with a PS3 media server) with the i7 desktop but am concerned about leaving it on all the time. What stripped down, low energy usage distro can I couple with which windows &amp; PS3 friendly software?  Also, how should i reformat my i7 (in terms of partitions, RAID options, whatever)? It has 2 1Tb drives (likely adding a third), and 1 320 gig. Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct? Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why? For US federal tax, whether a taxpayer is subject to AMT is determined by a complicated way in Instructions for Form 1040, Line 45, which involves an incomprehensible "worksheet to see if you should fill in form 6251".  By simplification and approximation, is there a rule of thumb for an estimate of a salary value of a taxpayer, so that a salary above the value might cause the taxpayer to pay AMT with a big possibility, and a salary below the value might not with a big possibility?  Why do the following two sources seem to differ greatly in estimates of the boundary salary value:    says that  the possible boundary salary value for a single taxpayer  to pay AMT is $53,900.  says that      Taxpayers with small incomes fall within the AMT exemption limits.   That leaves the upper middle class to face the tax, especially those   with incomes between $200,000 and $500,000    Thanks. I'm fairly new to DMing and so is my group, so we are all on a learning curve together. We're playing D&amp;D 5e.  Last week a fighter in our group tamed a wolf with Animal Handling and then requested that he keep the wolf. I allowed this, but now I'm not sure about how to play the wolf.  Specifically, how can he train the wolf to perform certain actions such as attacking?  What other things might I need to be aware of with a PC owning a "pet"? How to Search for Files Recursively into Subdirectories I am trying to look for all XML files in a particular directory and all sub-directories (recursively) inside it.  ls -R *.xml is only listing files in the current directory. I am quite sure, the sub-folders themselves have several .xml files, but none are showing up.  Is this a configuration issue? Declaring Text Decorations such as Underline, Strikethrough in a Style How do I include text decorations such as Underline, Strikethrough etc in a Style definition:  &lt;Style x:Key="UnderlinedLabel"&gt;      &lt;Setter Property="Control.FontFamily" Value="Trebuchet MS" /&gt;      &lt;Setter Property="Control.FontSize" Value="14" /&gt;      &lt;!-- Next line fails --&gt;      &lt;Setter Property="Control.TextDecorations" Value="Underline" /&gt; &lt;/Style&gt;   I'm familiar with using the following XAML to underline text:  &lt;TextBlock&gt;    &lt;Underline&gt;        Underlined text    &lt;/Underline&gt; &lt;/TextBlock&gt;   However text decoration is just another style, I want to be able to define it declaritively like FontWeight, FontSize etc.  [Update]  I was applying this style to a Label control.  This was my main problem.  It appears you can't underline text in a Label.  Change to a TextBlock (thanks gix) and all is well. When visiting my profile on the mobile app, my hat scrolls with the page:    It doesn't stay linked to my icon at the top. I'm on an iPhone 6+ running iOS 9.2 and version 1.4.1 of the SE app. Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$ Let $f$ be analytic on an open set $U$, let $a \in U$, and $f'(a)\neq 0$.  Show that  \begin{align*} \frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz \end{align*} where $C$ is a small circle centered at a.  I thought that this is very similar with Cauchy's integral formula. But, I can't apply the Cauchy's integral formula. How to apply that?   Any help is appreciated..   Thank you! How to plot logistic regression decision boundary? I am running logistic regression on a small dataset which looks like this:    After implementing gradient descent and the cost function, I am getting a 100% accuracy in the prediction stage, However I want to be sure that everything is in order so I am trying to plot the decision boundary line which separates the two datasets.  Below I present plots showing the cost function and theta parameters. As can be seen, currently I am printing the decision boundary line incorrectly.    Extracting data  clear all; close all; clc;  alpha = 0.01; num_iters = 1000;  %% Plotting data x1 = linspace(0,3,50); mqtrue = 5; cqtrue = 30; dat1 = mqtrue*x1+5*randn(1,50);  x2 = linspace(7,10,50); dat2 = mqtrue*x2 + (cqtrue + 5*randn(1,50));  x = [x1 x2]'; % X  subplot(2,2,1); dat = [dat1 dat2]'; % Y  scatter(x1, dat1); hold on; scatter(x2, dat2, '*'); hold on; classdata = (dat&gt;40);   Computing Cost, Gradient and plotting  %  Setup the data matrix appropriately, and add ones for the intercept term [m, n] = size(x);  % Add intercept term to x and X_test x = [ones(m, 1) x];  % Initialize fitting parameters theta = zeros(n + 1, 1); %initial_theta = [0.2; 0.2];  J_history = zeros(num_iters, 1);   plot_x = [min(x(:,2))-2,  max(x(:,2))+2]  for iter = 1:num_iters  % Compute and display initial cost and gradient     [cost, grad] = logistic_costFunction(theta, x, classdata);     theta = theta - alpha * grad;     J_history(iter) = cost;      fprintf('Iteration #%d - Cost = %d... \r\n',iter, cost);       subplot(2,2,2);     hold on; grid on;     plot(iter, J_history(iter), '.r');  title(sprintf('Plot of cost against number of iterations. Cost is %g',J_history(iter)));     xlabel('Iterations')     ylabel('MSE')     drawnow      subplot(2,2,3);     grid on;     plot3(theta(1), theta(2), J_history(iter),'o')     title(sprintf('Tita0 = %g, Tita1=%g', theta(1), theta(2)))     xlabel('Tita0')     ylabel('Tita1')     zlabel('Cost')     hold on;     drawnow      subplot(2,2,1);     grid on;         % Calculate the decision boundary line     plot_y = theta(2).*plot_x + theta(1);  % &lt;--- Boundary line      % Plot, and adjust axes for better viewing     plot(plot_x, plot_y)     hold on;     drawnow  end  fprintf('Cost at initial theta (zeros): %f\n', cost); fprintf('Gradient at initial theta (zeros): \n'); fprintf(' %f \n', grad);   The above code is implementing gradient descent correctly (I think) but I am still unable to show the boundary line plot. Any suggestions would be appreciated.    logistic_costFunction.m  function [J, grad] = logistic_costFunction(theta, X, y)      % Initialize some useful values     m = length(y); % number of training examples      grad = zeros(size(theta));      h = sigmoid(X * theta);     J = -(1 / m) * sum( (y .* log(h)) + ((1 - y) .* log(1 - h)) );      for i = 1 : size(theta, 1)         grad(i) = (1 / m) * sum( (h - y) .* X(:, i) );     end  end   EDIT:  As per the below answer by @Esmailian, now I have something like this:  [m, n] = size(x);  x1_class = [ones(m, 1) x1' dat1']; x2_class = [ones(m, 1) x2' dat2'];  x = [x1_class ; x2_class] do I only get a limited number of at-will powers?