I was looking for the proof is this theorem, but I couldn't find it anywhere.  the theorem is stated formally:  If $f_m$ is a sequence of continuous functions defined on $D$ (subset of $R$) such that $f_m$$\to$$f$ uniformly on $D$ then $f$ is continuous.  can someone give the stepwise proof?
The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all.
What's a command line way to find large files/directories to remove and free up space? Looking for a series of commands that will show me the largest files on a drive. Is it possible to have a different background for each workspace? I normally use 4 different workspaces for different things, and they are always the same. Can I make a different background image for each one of them?   I would like an answer that works for GNOME 3 desktop environment, Unity and the Classic desktop environment.     This question has multiple answers covering different Ubuntu versions: How to handle dynamic changing ID's In XPath? I see this a lot, someone in a category of people attempts to shut down anyone with an opinion that is not from the same category. For example, people without children weighing in on parenting; the parent may say something like "well, since you don't have kids and I do, my position is of more validity than yours."  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The scroll-bar gets stuck and makes it difficult to read the text.  I'm running Chrome Version 56.0.2924.87 (64-bit) on macos. how can I extrude curve shapes onto curve without tapering I'm stuck. Can't figure out how I can extrude a closed curve onto a NURBS curve without it looking tapered.   I notice that when it's set to "2D" there is no tapering, but I want to do it on 3d.  Thanks for help. Why is it that they decided to make String immutable in Java and .NET (and some other languages)? Why didn't they make it mutable? How can I know if a process is running? When I get a reference to a System.Diagnostics.Process, how can I know if a process is currently running? Water Wars: Time Travel TV Show? I have been playing for almost 2 days, and today it is asking me to sign up again.  I have all my game saved in my google account, but I wonder if I just log in from a trainer account will I keep my progress? How to deal with perfect separation in logistic regression? If you have a variable which perfectly separates zeroes and ones in target variable, R will yield the following "perfect or quasi perfect separation" warning message:  Warning message: glm.fit: fitted probabilities numerically 0 or 1 occurred    We still get the model but the coefficient estimates are inflated.   How do you deal with this in practice? How can I switch back to Nvidia card from Intel with Nvidia-prime So I updated to Ubuntu 14.04 and found that the primusrun command no longer works, so ditched bumblebee in favour of nvidia-prime. This lets you switch video card from nvidia-settings rather than running through the command line (I prefer the old method, but want the primus performance).  The problem is that after switching to Intel from nvidia-settings, I can no longer open it up again to switch back to the nvidia card. When I open nvidia-settings I get a segmentation fault (according to the terminal).  Is there any way to open nvidia-settings switch back? I would like the ability to be able to switch between the two since having the nvidia card on all the time kills my battery and heats up the laptop a lot. How do I format a number in Java? Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z[\frac {1}{2}]$ Rename ToC, List of Figures, References I would like to know how to rename the standard list of figures, table of contents, and references. What I saw so far relying on \renewcommand{\contentsname}{name} did not work so far.  Here's the list of preambles I use:  \documentclass[12pt]{report} \usepackage[utf8]{inputenc}  \usepackage{fourier} \pagestyle{headings} \usepackage{marvosym} \usepackage[pdftex]{graphicx} \usepackage[margin=2.5cm]{geometry} \usepackage{rotating} \usepackage{amsmath} \usepackage{booktabs} \usepackage{url} \usepackage{natbib} \usepackage{setspace} \linespread{1.5} \newenvironment{remark}{\par\mbox{}\par\noindent{\bfseries             \emph{Remark}\/}:\normalfont}{\par} \newenvironment{defin}{\par\mbox{}\par\noindent\bfseries \emph{\Pointinghand\     }\normalfont}{\par} \setcounter{toc**strong text**depth}{2}  \usepackage[colorlinks=true, a4paper=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} \pdfcompresslevel=9 \setstretch{1.0} %vertical stretch \usepackage{fontspec} \setmainfont[BoldFont       = AvenirNextRoundedPro-Bold.otf , ItalicFont     =AvenirNextRoundedPro-Italic.otf]{AvenirNextRoundedPro-Reg.otf} \defaultfontfeatures{Mapping=tex-text} \usepackage{multirow} \usepackage{array} \usepackage{booktabs} \usepackage{hyperref} \hypersetup{ hidelinks, } \usepackage{ragged2e} \usepackage{euler} \usepackage[euler-digits,euler-hat-accent]{eulervm} \usepackage[labelfont=bf]{caption} \usepackage{wrapfig} \usepackage{amsmath} \usepackage{amssymb}  \usepackage[english]{babel} \usepackage[nottoc]{tocbibind} %Adds "References" to toc \usepackage{titlesec} \titleformat{\chapter}[display]{\bfseries\huge}{\filleft\Large\chaptertitlename~\thechapter}{1ex}{\vspace{1ex}\filright}[\vspace{0.5ex}\titlerule] \usepackage{float} \usepackage{etoolbox} \makeatletter \patchcmd{\@caption}{\csname the#1\endcsname}{\csname fnum@#1\endcsname:}{}{} \renewcommand*\l@figure{\@dottedtocline{1}{2em}{6em}} \let\l@table\l@figure  \makeatother % default for 3rd arg: 2.3em - for list of figures and tables \usepackage{background} \definecolor{textcolor}{HTML}{000000}             \newcommand\Text{c\quad o\quad n\quad f\quad i\quad d\quad e\quad n\quad t\quad i\quad a\quad l} \SetBgColor{textcolor}\SetBgOpacity{0.3}\SetBgAngle{90}         \SetBgPosition{current page.center} \SetBgVshift{-0.33\textwidth} \SetBgScale{1.8}         \SetBgContents{\sffamily\Text} %for confidential   I also use TexMaker. I do not know how relevant it is, but here my document language is set to English (even if it's not the language I write in, in that document, but it isn't a problem seemingly). What I also saw is that, if I remove babel and the references, I can rename my ToC and LoF, LoT. But I need the references though. Any tip would be very welcome!  Edit: I cannot provide text data from my project, because it is confidential. Here's what I can provide from the main document. Here is what I can include more.  \begin{document} \include{02_front}  \setcounter{page}{0} \pagenumbering{roman} \tableofcontents \listoffigures \listoftables \newpage \setcounter{page}{0} \pagenumbering{arabic} \chapter{blabla} \section{test é è åøæ î} \appendix \bibliographystyle{plain}  \bibliography{80_refs}   \end{document}   Also, I actually finished writing my thesis but I need to summarize it briefly in another language, am would be extremely unwilling to test and remove packages that "might be" not very useful. I literally only have to change these names of ToC, LoF, ToF and references. $24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement. Which are the spec bits in this cache? I have a direct mapped cache of size S with the line size L. The cache is physically indexed and tagged. The physical address is 50 bits, numbered from 0 to 49 (with 0 being the least significant bit). The machine has a word size of 4 bits and the memory is byte-addressable. Which bits of the 50 bits are compared with the address tag? When to use LinkedList over ArrayList in Java? I've always been one to simply use:  List&lt;String&gt; names = new ArrayList&lt;&gt;();   I use the interface as the type name for portability, so that when I ask questions such as these I can rework my code.    When should  be used over  and vice-versa?