Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function with $\lim_{x \rightarrow + \infty}f(x)=0$ and $\lim_{x \rightarrow - \infty}f(x)=0$. Show that $f$ is uniformly continuous on $\mathbb{R}$  My attempt: Take a random $\epsilon &gt;0$. As  $\lim_{x \rightarrow + \infty}f(x)=0$, there exists a $M_{1} \in \mathbb{R}$ so that for every $x \in \mathbb{R}$ with $x &gt; M_{1}$ it holds that $|f(x)-0|&lt; \epsilon$. As  $\lim_{x \rightarrow - \infty}f(x)=0$, there exists a $M_{2} \in \mathbb{R}$ so that for every $x \in \mathbb{R}$ with $x &lt; M_{2}$ it holds that $|f(x)-0|&lt; \epsilon$. Now take $M \geq \max \{|M_{1}|,|M_{2}|\}$. Then surely will $|f(x)|&lt; \epsilon$ for every $x \in \mathbb{R} \backslash [-M,M]$.  Now I'm not sure how to go on from this. I do know that $f$ is uniformly continuous on the interval $[-M,M]$ because it is continuous. But how can I expand this to $]- \infty,+\infty[$ ?
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty$. Prove that $f$ is uniformly continuous.   Since $f(x)\to 0$ as $x\to \pm\infty$ we have $|f(x)|&lt;\epsilon $ whenever $x&lt;-K$ and $x&gt;K$ where $K&gt;0$.  Also $f$ is uniformly continuous on $[-K,K]$ as the domain is compact.  Hence $f$ is uniformly continuous in $(-\infty,-K)\cup [-K,K]\cup (K,\infty)$.  But how to conclude that $f$ is uniformly continuous on $\Bbb R$ from above as I may chose $x\in (-\infty,-K);y\in [-K,K]$ ,how to conclude that $|f(x)-f(y)|&lt;\epsilon $ from above for a given $\epsilon$?  Please suggest the edits required.
How does one get the MinionPro \qoppa? Since I have switched to the 2014 texlive, one issue is not working: the character \qoppa. The command is defined in polutonikogreek in babel. It is an archaic greek letter, but was used also in classic times for the number 90.  The glyph is available in the cmr font and in MinionPro and in the beautiful MinionMath font of Johannes Küster. I want the latter. In MinionPro/MinionMath, it is unicode character 03d9 (formerly u+03DE, it recently changed, as it turns out), see   How do I get the glyph into latex? I tried  \def\testqoppa{\begingroup\fontfamily{MinionPro}\fontencoding{LGR}\selectfont\char19\endgroup}   and   \def\testqoppab{\begingroup\fontfamily{minionmath}\fontencoding{LGR}\selectfont\char19\endgroup}   But all this gives the cmr glyph, because latex complains  LaTeX Font Warning: Font shape `LGR/MinionPro/m/n' undefined LaTeX Font Warning: Font shape `LGR/minionmath/m/n' undefined (Font)              using `LGR/cmr/m/n' instead on input line 571.   This seems like a beginner error from my side. Can somebody help me to get it right? I need \qoppa only in two places, so a nasty trick is also sufficient. The MinionMath font is installed and is working for all other purposes. Here is an extract of the .cls file:  \usepackage[fullfamily,textlf, opticals, swash, onlytext]{MinionPro} \usepackage[extraops,textcomp,withamsmath,amssymb]{minionmath} \usepackage{amsmath}             \usepackage{minionamsmath} \usepackage[polutonikogreek,greek,latin,french,italian,german,british]{babel} Ellipse formula from points Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)?  Background: Using R Momocs library function  which returns confidence ellipse coordinates. Would be nice to abstract those points into a formula.  You can see my ellipse points in the image below;    Any suggestions much appreciated, Thanks What would you call a person who doesn't want to learn anything new? Why is the year NOT displayed in the questions and comments dates? Why is it that the year is never displayed in the questions or comments? Is it because it is the current year? I, and maybe others, wonder. Order of some quotient ring of Gaussian integers Energy conservation in General Relativity I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system?  Here are some examples, is there a convenient way to define energy in these scenarios?   Just a system of gravitational waves. A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move. Two rotating bodies of similar mass.   Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. Why can't we do the same when we break TT symmetry by making space-time curved? I have a question.  I have been given this proof: "For any $n$ in the integers where $n&gt;2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$."  I have gone through and actually proved this, (that the numbers are $1$ and $n-1$) but i didn't' know how to prove that $n-1$ is in fact in the set $U(n)$. Is it because two consecutive numbers are always relatively prime? How do I run .sh scripts? Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? How do you get the index of the current iteration of a foreach loop? Is there some rare language construct I haven't encountered (like the few I've learned recently, some on Stack Overflow) in C# to get a value representing the current iteration of a foreach loop?  For instance, I currently do something like this depending on the circumstances:  int i = 0; foreach (Object o in collection) {     // ...     i++; } Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ Calculate  $$\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx$$   I have tried to put $\displaystyle x=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$  $$ \int^{\infty}_{0}\frac{t^2\ln^2(t)}{(t^2-1)^2}dt$$  $$\frac{1}{2}\int^{\infty}_{0}t\ln^2(t)\frac{2t}{(t^2-1)^2}dt$$  $$ \frac{1}{2}\bigg[-t\ln^2(t)\frac{1}{t^2-1}+\int^{\infty}_{0}\frac{\ln^2(t)}{t^2-1}+2\int^{\infty}_{0}\frac{\ln(t)}{t^2-1}dt\bigg]$$  How can I solve it? In the Upcoming Halloween Update Posted By SOOPUR SELL (supercell), there have been sneak peeks released showing new skeleton traps for TH 8+. I also remember the Santa Strike from one of the Christmas Updates. My question is when these holiday events finish, do we lose the money we spent on the Traps or do we keep them forever? How to use "text" as a verb In Finder when I type something in the textbox the default is to search the whole mac in the contents of the files. I was able to change the default to search the current directory, but didn't see an option to change search for contents to search for file name. How can that be changed? Integral of Sinc Function Squared Over The Real Line I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks.  Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anyways. In Java Edition, I can name a mob by using NBT tags like this: /summon zombie ~ ~ ~ {CustomName:'{&quot;text&quot;:&quot;TheUndead&quot;}'}  But in Bedrock Edition, I've heard NBT is not accessible from commands. What can I do to summon a mob with a custom name tag using the /summon command? Suppose that $X_1,X_2,\ldots,X_n$ is a random sample from a distribution with probability function:  $$p_X(x)=\begin{cases} 1/2 &amp;x=-1,1 \\ 0 &amp; \text{otherwise} \end{cases}$$  Now if we define the average   $\bar{X}= \frac{1}{n} \sum X_i$, I need to show that that for an odd $n$ the probability function for $\bar{X}$ is:  $$p_{\bar{X}} (x)= \frac{\binom{n}{\frac{n}{2} (x+1)}}{2^n}$$ for $x=\pm 1/n,\pm 3/n,\ldots,\pm 1$, $0$ otherwise.    Since $P[\sum X_i=k] =P [\bar{X}=k/n ]$, I thought it would be easier to first derive the probability function for the sum. Since they are still $2^n$ n-tuples and each one is equilikely, I need to count the ones whose sum is $k$ for:  $k=\pm 1,\pm 3,\ldots \pm n-2,\pm n$ (since even numbers are not a feasible combination)  The problem here though is that I do not immediately recognize how I could count all these tuples. Could you please help me with this? If it is easier to proceed another way, I am of course all ears. How do I install python imaging library (PIL)? How do I install the python imaging library (PIL) on Ubuntu?  I tried doing  sudo apt-get install python-imaging   but now when i run     python selftest.py (a script I got from somewhere on the  website) I get (among other warning messages):  ... *** JPEG support not installed *** ZLIB (PNG/ZIP) support not installed ... *** 1 tests of 57 failed.   Have I somehow messed up the PIL -- how do I fix that?  Is maybe the PIL just fine, but that "selftest.py" is not really the right program for checking to see if PIL is installed properly -- how else can I tell if PIL is installed properly or not?  (I'm using Ubuntu 12.04 LTS "Precise Pangolin").  (What I'm ultimately trying to do is to add a 2D barcode generator to some python code, and all the 2D barcode generators I could find that were written in python all seemed to use the PIL). I'm preparing a document using the book class. In the table of contents, the space between the subsection number and title is too wide. How to reduce it? How to select (highlight) text in Skype chat without using a mouse? Recently I started learning to use keyboard shortcuts almost for everything I'm doing on the laptop, but soon enough after I started, I bumped into the problem of being unable to highlight (select) simple text in Skype chats.  I did some research and I couldn't find any solution to my problem. Have I missed an already answered question that is the same like mine or if not what solution can you suggest in order to achieve the results I want without touching mouse or touchpad or any cursor moving device? How to smooth shade an object while retaining hard edges? I frequently come to the problem of having a mesh with parts I would like to render smooth and "hard" edges connecting them.   How can I set the mesh to show the sharp edges but render everything else smooth?  In the example I would like the sides of the individual cylinders render smooth but the intersection between bottom and top to be sharp/flat