I want to prove that if $f$ is continuous at $c$ and $f(c)&lt;y$, then $\exists \delta &gt;0$ such that $\forall x \in Dom(f)$ and $x \in (c- \delta , c + \delta)$ implies $f(x) &lt; y$.   I know that $f$ is continuous at $c$ and $f(c) &lt; y$. So I should choose a $\delta$. But I am not sure how to progress after that.
Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) &gt; 0$ then there exists a $d &gt; 0$ such that $\forall x$, $f(x) &gt; 0$ with $|x-c| &lt; d$.   I can understand what the problem means. That if a function is positive at a point then there is another point that's really close that'll also be positive. I can not prove this in a formal way though. I've tried using the intermediate value theorem and I do not know hot to implement it. Any ideas?
Finding the ID of a process and killing it When ever I need to kill a background process I do ps -e | grep &lt;process_name&gt;  Which prints something like this 1766 ? 00:00:13 conky , Then I use the process ID to kill it like so kill 1766 .   Is there any way I can simplify this ? Make it quicker ? reduce the amount of typing ? Is “might could” a correct construct? Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$ I am trying to prove the summation formula using induction:  $$\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$$  So far I have...  Base case:   Let n=1 and test   $\frac{1}{k(k+1)} = 1-\frac{1}{n+1}$  $\frac{1}{1(1+1)} = 1-\frac{1}{1+1}$    $\frac{1}{2} = \frac{1}{2}$   True for n=1   Induction Hypothesis:   Assume the statement is true for the n-th case   $\sum_{k=1}^n  \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$  Inductive Step:   Prove, using the Inductive Hypothesis as a premise, that   $$\sum_{k=1}^{n+1}  \frac{1}{k(k+1)} = \sum_{k=1}^{n}  \frac{1}{k(k+1)} + \frac{1}{(n+1)(n+2)} = 1-\frac1{n+1} + \frac{1}{(n+1)(n+2)} = \frac{(n+1)(n+2)}{(n+1)(n+2)}+\frac{-2-n}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)} = \frac{(n+1)(n+2)-2-n+1)}{(n+1)(n+2)} = \frac{(n+1)(n+2)-n-1}{(n+1)(n+2)} = \frac{n^2+2n+1}{(n+1)(n+2)} = \frac{(n+1)(n+1)}{(n+1)(n+2)} = \frac{n+1}{n+2}$$  To prove $$ 1-\frac{1}{n+2} = \frac{n+1}{n+2} $$ Multiply both sides by $n+2$ to get an equivalent expression. $$ (1-\frac{1}{n+2}) * (n+2) = (\frac{n+1}{n+2}) * (n+2)  $$ $$ n+1=n+2−1 $$  Does this all make sense? How can this be improved upon? Are there any tools that I can use to defrag my Windows partition from Linux? How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$? Are the floor functions of $0.999\cdots$ and 1 equal?  It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without using $0.999\cdots=1$ how can we show that  $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$ ? How can I stop certain apps from appearing in the Dash? Addressing a former office-holder by that office's title When is it appropriate to use an "expired" honorific to address or refer to a person?   In the U.S., former state governors are occasionally referred to as "Governor So-and-so", although they have not held that office for several years. I see this happening most often when the former governor is actively campaigning for a new office (usually U.S. President), and my impression is that it's done only by someone with an interest in the campaign (for or against) rather than a (relatively) neutral party like a newspaper.  Certainly this could be used as a jab: in a political campaign for high office, addressing one's opponent by a former title — lower than one's own — could imply that the opponent is not qualified for the new job. It could also be used the other way around, to remind listeners that a candidate one supports used to hold an important title.  I'm curious whether there are formal rules for this in the U.S., or at least style or journalist guidelines, for various offices. Are ambassadors, for example, entitled to some sort of emeritus honorific? Using "you" and "your" as a representation for yourself and everyone in general Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. I believe it's a common practice in some communities to name children after their grandparents or parents or relatives and sometimes even friends and popular people. Places, roads, streets etc., get named after famous people, too. Many inventions and discoveries have been named after people who invented or discovered them.  But I am not yet aware of a term that refers to the person whose name is given to people, places or objects this way. Is there a suitable word to fill in the blank below?     They named their son after the famous football player, David Beckham.      David Beckham is .......... of their son. What are the requirements to use iMac screen as a second monitor - Target Display Mode? I have a Surface Pro 4 and an iMac (late 2019). I would like to connect my Surface Pro to the iMac to use the iMac screen as a second monitor.  Is that possible?  What are the requirements to use an iMac as a second screen? How do I create a script file for terminal commands? In Windows I can write a file containing commands for cmd (usually .cmd or .bat files).  When I click on those files it will open cmd.exe and run them commands the file contains.  How would I do this in Ubuntu?         I'm sure this is a duplicate, but I can't find my answer. Its similar to these questions, but they don't answer the question: Sequences in $C(X)$ converging uniformly implies uniform boundedness and equicontinuity Let $X$ be a metric space and let $\{ f_n \}_{n \geq 1}$ be a sequence in $C(X)$. Show that if $\{ f_n \}_{n \geq 1}$ converges uniformly on $X$ then $\{ f_n \}_{n \geq 1}$ is both uniformly bounded and equicontinuous.  Could I get a hint? If $a$, $b$, $c$ are the roots of $x^3-6x^2+3x+1=0$, find all possible values of $a^2b+b^2c+c^2a$ Let $a$, $b$, $c$ be the roots of $$x^3 - 6x^2 + 3x + 1 = 0$$ Find all possible values of $a^2 b + b^2 c + c^2 a$. Express all the possible values, in commas.  I've already tried to bash out all the roots, Vieta's Formula, and try to manipulate the equation to become easier to work with. However, Vieta's didn't get me anywhere, and I couldn't find a way to make the equation simpler or anything. Any hints to start this problem? In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. Why there is a delay between when a key is pressed and when it is accepted? This is one of the wackiest and weirdest problems I have ever encountered. My keyboard was working absolutely fine untill yesterday on Ubuntu 13.04. However, when I boot to Ubuntu today, I suddenly find that all the keys work, but I have to press and keep on holding them for about a second for them to work.   This behavior is true for all the keys except for Num Lock, Caps Lock and Scroll Lock keys, the indicator for them on my keyboard lights up as soon as I press those keys. The keyboard works fine at the login screen where I put my password.  The keyboard also works normally if I use the Guest session instead of my normal user account. I have not done any key-remappings and using the standard English (US) keyboard layout.  I have fiddled around with the settings for Keyboard in System Settings, but to no avail. I have Windows 7 as dual-boot and the keyboard works perfectly well on it.   Why am I facing such a behavior and how to normalize this?  PS: Using Logitech Classic Keyboard K100 (USB Model). Turning fullscreen on made Minecraft crash, and now it crashes on start every time I switched to full screen on minecraft, the screen turned black as if it were loading it took a while so i closed minecraft and reopened it. When I logged on it gave me a program on my taskbar saying "minecraft minecraft" then when i closed the program it gave me this error message:   java(tm) platform SE binary has stopped working.    Is there a way to fix this? Why do some tag info pages show who created them and others don't? Some  show who created the tag:    while  don't. I'm pretty sure I created that one as well, if you hover over the timestamp you can see when it was created:    which more or less matches with  where I created the tag:    I understand that some tags like  were probably created by the Community user while the site itself was created (so it ), but some newer tags like   even though  still has an account here.  What's going on here? Would this also mean I wouldn't get a  for  if I ? How to set a font for math mode? I would like to have something like Segoe UI of something similar. Thank you  \documentclass{article} \usepackage{fontspec} \usepackage{firamath-otf} \begin{document} \begin{equation} x+y=45 \end{equation}  \end{document}   Error: Diagonal straight lines in residuals vs fitted values plot for multiple regression I am observing strange patterns in residuals for my data:   [EDIT] Here are the partial regression plots for the two variables:      [EDIT2] Added the PP Plot   The distribution  seems to be doing fine (see below) but I have no clue where this straight line might be coming from. Any ideas?   [UPDATE 31.07]  It turns out you were absolutely right, I had cases where the retweet count was indeed 0 and these ~ 15 cases resulted in those strange residual patterns.   The residuals look much better now:   I've also included the partial regressions with a loess line. An ideal whose radical is maximal is primary I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n&gt;0$.    I've been trying for a while and I'm not sure where to begin. All I've got is that if $\text{Rad}(Q)$ is maximal and by definition it's the intersection of all prime ideals $P_i$ containing $Q$, then it must be equal to each of these $P_i$. So there is only 1 prime ideal containing $Q$, namely $\text{Rad}(Q)$.  Could anyone point me in the right direction? Thanks for any replies.