Prove that $f: [a,b]\to \mathbb R$, $|f(x)-f(y)| \leq 4|x-y|$ is integrable, when $f$ reaches its maximum in any closed subinterval. Let $f:[a,b]\to \mathbb R$ such that it reaches its maximum and minimum in any closed sub-interval of $[a,b]$.  In addition:  $$|f(x)-f(y)| \leq 4|x-y|$$  holds for any $x,y$ in the interval. How would you prove that the function is integrable using the fact that Riemann's condition (i.e. $U-L &lt; \epsilon$ for any epsilon)?
Prove Lipschitz function $f$ with constant $K$ is integrable on $[0, 1].$ We suppose that $f : [0, 1] \rightarrow \mathbb{R}$ is a Lipschitz function with constant $K$. We want to show that $f$ is integrable on $[0, 1].$  I've been trying to use the Darboux criterion of integrability by trying to show that, for $\epsilon &gt; 0$, $$S^*(f,P) - S_*(f, P) &lt; \epsilon,$$ where $S^*(f, P)$ and $S_*(f, P)$ are the Upper and Lower Riemann Sums, respectively.  This is what I have so far: $$\sum_{i=1}^n M_i (x_i - x_{i-1})- \sum_{i=1}^n m_i (x_i - x_{i-1}) = \sum_{i=1}^n (M_i - m_i) (x_i - x_{i-1})$$ $$\le \sum_{i=1}^n K (x_i - x_{i-1}).$$ because $|f(x) - f(u)| \le K|x-u| \le K |1|$, since $x\in [0, 1]$. Then, $$\sum_{i=1}^n K (x_i - x_{i-1}) = K.$$ However, this is only a single upper bound, not an arbitrary $\epsilon.$ Can I get some pointers about how to proceed? Thanks!
Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} $$ For any $A,B$, I want to prove that $d(A,B) = d( \overline{A}, \overline{B} )$. My Attempt Put $\alpha = d(A,B)$. Therefore, $\alpha = \|x_0 - y_0 \|$ for some $x_0 \in A$ and some $y_0 \in B$. But, notice $x_0 \in \overline{A}$ and $y_0 \in \overline{B}$ by definition. hence $$ \| x_0 - y_0\| \geq \inf\{ \|x' - y'\| : x' \in \overline{A}, \; \; y' \in \overline{B} \} =  d( \overline{A}, \overline{B} )$$ $$\therefore d(A,B) \geq d( \overline{A}, \overline{B} )$$ I am stuck trying to show the other direction: $d(A,B) \leq d( \overline{A}, \overline{B} )$ Can someone help me? thanks a lot How to recover deleted files? Are there any tools, methods, incantations to recover recently deleted files on Ubuntu?  If it makes any difference, I want to recover a  2.x database file. But would be better to have a method/tool that works on any kind of file. How do I bypass restrictions on the length of the PATH variable System Properties      This environment variable is too large. This dialog allows setting values up to 2047 characters long.   I use/have used a lot of software, and my PATH has grown as a result. When I try to add new paths, I get the above message. As a workaround, I'm editing my user path, but prefer not to. Internal keyboard and trackpad stop working after plugging in USB device I got a new MacBook Pro with OS X Mavericks. When connecting my phone via USB, the internal keyboard and the trackpad immediately stop working -- not always but in most cases. The bluetooth mouse and an external USB keyboard are not affected. Everything else works also fine. The power button of the internal keyboard still works. Reboot helps. Any ideas? Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected? Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon &gt;0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)&gt; \varepsilon \}. $$ Then  $U_\varepsilon$ is open but in general not connected.  But I suppose that for sufficiently small $\varepsilon &gt;0$ also  $U_\varepsilon$ is connected. How to prove its?  Edit. Let's assume that additionally $U$ is bounded. Is then $U_\varepsilon$ connected? What is the Fastest a Character Can Move in One Turn? At level 20 with any magic items in the DMG, and any spells, what is the fastest a DnD 5e character could move using their movement?  (i.e., not including teleportation) Is there a way to make IE8 honour opacity on an `:before` pseudo element? How should I print types like off_t and size_t? How are embedded quotations used? How would embedded quotations be used when quoting from passages/sources within an essay? What would be the difference between normal quotations and embedded quotations? \end{abstract} conflicts with arabtext package I am using elsarticle  Everything was find before I used \usepackage{arabtext}.   When I compiled to PDF, I got the following error message.    And if I type "s", everything became fine again but I don't want to type an "s" every time I compile PDF preview.   Please anyone advise.  Thank you very much in advance for your help.  Here is the code  \documentclass[12pt, authoryear]{elsarticle}    % use "amsart" instead of "article" for AMSLaTeX format \usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} \usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots. \geometry{a4paper}                          % ... or a4paper or a5paper or ...  %\geometry{landscape}                       % Activate for rotated page geometry %\usepackage[parfill]{parskip}          % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx}               % Use pdf, png, jpg, or epsÂ§ with pdflatex; use eps in DVI mode                             % TeX will automatically convert eps --&gt; pdf in pdflatex         \usepackage{amssymb}  \usepackage{amsmath} \usepackage{tipa} \usepackage{arabtex} %SetFonts  %SetFonts   \title{Translating from English to Bahasa Indonesia: Which procedures should be taught to EFL students preparing to be a translator?} \author[mkl]{Mohammad Kholid}  \author[fai]{Faisal Mustafa\corref{cor1}} \ead{faisal.mustafa@unsyiah.ac.id}  \cortext[cor1]{Corresponding author}  \address[mkl]{Syiah Kuala University, Banda Aceh, Indonesia} \address[fai]{Syiah Kuala University, Banda Aceh, Indonesia} \date{}                         % Activate to display a given date or no date  \begin{document} %\section{} %\subsection{}  \begin{frontmatter}  \begin{abstract}   Abstract content is here.  \end{abstract}  \begin{keyword}  translation procedures \sep English to Bahasa Indonesia \sep teaching translation \end{keyword} \end{frontmatter} Proof of Divergence for a Sequence Show that $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k. I think I have it figured and would like to see if I am on the right track. Thank you. I would like to be able to transpose a tabular(x) table like in the following example. It is not necessary to be completely automated, but I don't want to change the data order, which is a painful and stupid. By "not completely automated" I mean that e.g. the headers can be done seperately etc.  Let's see the MWE:  \documentclass{article}  \usepackage{booktabs} \def\Midrule{\midrule[\heavyrulewidth]}  \begin{document}  \centering  \begin{tabular}{*6c}\toprule \bfseries Name &amp; Alice &amp; Bob &amp; Chuck &amp; Dave &amp; Eve\\\Midrule \bfseries Sex &amp; Female &amp; Male &amp; Male &amp; Male &amp; Female\\\midrule \bfseries Age &amp; 18 &amp; 19 &amp; 20 &amp; 21 &amp; 22\\\bottomrule \end{tabular}  \bigskip  Which I want to look like  \bigskip  \begin{tabular}{*3c}\toprule \bfseries Name &amp; \bfseries Sex &amp; \bfseries Age\\\Midrule Alice &amp; Female &amp; 18\\\midrule Bob &amp; Male &amp; 19\\\midrule Chuck &amp; Male &amp; 20\\\midrule Dave &amp; Male &amp; 21\\\midrule Eve &amp; Female &amp; 22\\\bottomrule \end{tabular}   \end{document}     Note: I know that , but I would like to get it without expl3 (mostly because people would might use my code quite often have an old LaTeX installed and I cannot be convincing them to change this). How do I fix Google Chrome's latest new tab page? I've just updated to the latest Google Chrome Dev build, and they've changed from the annoying-but-ok new tab page to an ugly and less usable page.  On the old one, nine thumbnails was not enough, and now they've restricted it to eight.  There's an empty "Tips and Suggestions" box which I wouldn't want even if it had stuff in; they've added Recent Downloads - on the rare occasions when I need that, I'll hit Ctrl-J to open the Downloads tab; and the site-specific search box on the right has now gone!  Is there a way to get the old new tab page back, or a plugin to provide a superior one?   Update: I launch Chrome by pressing Win+R and entering "chrome" - I have a shortcut on the Path with a Target of "C:\Documents and Settings\&lt;user&gt;\Local Settings\Application Data\Google\Chrome\Application\chrome.exe" --old-new-tab-page and whilst double-clicking this shortcut brings up the previous new tab page, using the keyboard method results in the old one.  How do I get it working via the Run box? (Without using a batch file, which slows things down too much.)  Update 2: GRRR! Now they've gone and stopped the --old-new-tab-page parameter from working at all! I hate this new version of the page that they're forcing upon us - can anyone provide a method to get rid of it and get the decent page back? Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist Is there a way to make bash more TAB friendly? I use bash on both my Mac and as my cygwin shell. Overall, I'm quite pleased with it.  My one complaint however is that its TAB completion isn't as good as Windows Command Line TAB completion.   For example, if there are two files with the same prefix, then bash just beeps at me when I try to TAB complete. If I keep nailing TAB, then it will eventually show me a list of potential files.  I definately prefer how Windows will cycle through potential files right on the command line.   Is there a way to get bash to operate similar to Windows Command Line in this regard?  If there isn't (I do not want to switch shells), am I missing something about TAB completion in bash? Is there a more efficient way to use it? I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that $X\sim$Uniform$[0,1]$ and $Y\sim$Uniform$[0,1]$. My professor wrote:      $f_{X+Y}(Z) = \int_0^Z f_X(x) f_Y(Z-x)\;dx \neq Z$   Then somehow he came up with these cases where $Z$ is defined, and used that in making the pdf. I'm just not sure of how he did this or how he also came to find $P(X_1+X_2\le 2)$. Can someone help me fill in the blanks? I went to him directly but he just said the same thing as in lecture and wasn't helpful. Mount error: "unknown filesystem type 'exfat'" When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? For simple linear regression, the regression coefficient is calculable directly from the variance-covariance matrix $C$, by  $$ C_{d, e}\over C_{e,e} $$ where $d$ is the dependent variable's index, and $e$ is the explanatory variable's index.  If one only has the covariance matrix, is it possible to calculate the coefficients for a model with multiple explanatory variables?   ETA: For two explanatory variables, it appears that $$\beta_1 = \frac{Cov(y,x_1)var(x_2) - Cov(y,x_2)Cov(x_1,x_2)}{var(x_1)var(x_2) - Cov(x_1,x_2)^2} $$ and analogously for $\beta_2$. I'm not immediately seeing how to extend this to three or more variables. Changing order of features in shapefile