When $n=0$  $1&gt;\frac{1}{2}$   (true)  Then I prove that  $$2×n!\geq2^n$$ And $$(n+1)\times n!\geq2^n$$ So I have to prove that $n+1\geq2$  $n\geq 0$  $n+1\geq 1$ (not $2$). Could someone point out what is wrong with my reasoning please?
I'm having difficulty solving an exercise in my course.  The question is:      Prove that $n!\geq 2^n$.   We have to do this by induction. I started like this:   The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$.   Now proof for $(n+1)$ which brings me to: $(n+1)! \ge 2^{(n+1)}$  I think I can rewrite it somehow like this:  $$ {(n+1)} \times {n!}  \stackrel{\text{(definition of factorial)}}{\ge} 2^n \times 2 $$  $$ (n+1) \times 2^n \ge 2^n \times 2 $$  Then I think I can eliminate the $2^n$ and have something like this: $n+1 \ge 2$, or $n \ge 1$.  But I think I'm wrong here some where and was hoping somebody has some advice on this. How can I prove the above assumption?  Any help would be appreciated, kind regards.
"Year olds" or "year-olds" I am writing about some leadership stuff, and I am trying to say that I am leading the 14-15 year old boys. How do I correctly use year-olds as in, "I was the adviser to the 14-15 year-olds."  Thanks in advance /usr/sbin/grub never finishes I did apt-get upgrade it never finishes at this point:  Searching for GRUB installation directory ... found: /boot/grub Searching for default file ... found: /boot/grub/default Testing for an existing GRUB menu.lst file ... found: /boot/grub/menu.lst Searching for splash image ... none found, skipping ... Found kernel: /boot/vmlinuz-3.2.0-24-virtual Updating /boot/grub/menu.lst ... done  You already have a NoLOADER configuration in /etc/grub.conf Running boot loader as requested Running /usr/sbin/grub  ...    The system does not freeze (Ctrl-Z works, I get my console back) just that that /usr/sbin/grub seems to never complete.  What should I do? Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$ Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact answer by writing $f(z)$ as $u(x,y)+iv(x,y)$ and realizing the Laplacian of f is equal to zero. This leads to a lot of cancellations but I'm still making mistakes in my computations and I can't figure out where. Any help is appreciated. What effect does serial downvoting have on my rep cap? Proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$ Prove that      $$ \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx = \ln\left(\frac{a}{b}\right) $$   My Attempt:  Define the function $I(a,b)$ as  $$ I(a,b) = \int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}\,dx $$  Differentiate both side with respect to $a$ to get  $$ \begin{align} \frac{dI(a,b)}{da} &amp;= \int_{0}^{\infty}\frac{0-e^{-ax}(-x)}{x}\,dx\\ &amp;= \int_{0}^{\infty}e^{-ax}\,dx\\ &amp;= -\frac{1}{a}(0-1)\\ &amp;= \frac{1}{a} \end{align} $$  How can I complete the proof from here? Contraction of quotient ideal is quotient of contractions? Monotonicity in argmin function I have the following equation $$f(a)=\text{argmin}_{t\in\mathbb{R}} \sum_{i=1}^n\frac{1}{2}(1-a)(b_i-t)^2+a| b_i-t |, a\in[0,1]$$  How does one prove that the function is monotone in a? Execute task if tasker profile has been active for some time How can I execute a tasker task only after a certain condition has been true for a certain time?  Normally a profile executes its task as soon as all conditions are met. I want to delay this activation, until the conditions have stayed true for some time.  E.g.    do something after the screen has been continuously off or on for 10 minutes do something after wifi xy has not been near for 10 minutes do something after cell towers x,y,z have not been near for 10 minutes. How to tell how many CPUs (cores) you have on Windows 7 One can contract I have to I've when have is a helping verb, e.g.     I've got an octopus in my pants.   Is contracting the main verb technically incorrect or merely antiquated? My father loves to say,     I've a month-old smoked shoulder I can cook for dinner,   and he sounds like a crazy old man when he does. I would like to know if I can shut him down by informing him that he is incorrect. :) How to create a virtual audio device and stream audio input with it Here's my concrete problem:  I am broadcasting a game via XFire and it uses the Windows audio device to capture any audio I receive. As I am broadcasting, other users who watch the video stream are communicating with me over Skype, and they hear themselves back within the video stream and it is entirely logical since I am broadcasting the audio I hear.  What I want to do is create another audio device within Windows and redirect (pipe) ONLY the audio input from that game and not the input reveived from Skype. I would then tell XFire to use that newly created "virtual" audio device to broadcast and therefore my partners won't hear themselves back.  Is it possible to create another audio device and redirect only wanted input streams to it? 16.04 new installation gives grub-efi-amd64-signed failed installation /target/ ubuntu 16.04 at the end I am installing Ubuntu 16.04 in an Intel NUC computer with UEFI enabled. I am using an SSD from another computer. During the installation I chose Erase the disk and make a new full install  At the end of the process I see:  grub-efi-amd64-signed failed installation /target/ Ubuntu 16.04    And the system does not boot.  I tried to use boot-repair but apparently it does not solve the problem. What is sudo apt-get -f install? what is the usage of apt-get -f install . when I run sudo apt-get update I got some errors at the end It says try apt-get -f install. What is this command?  the last part of my error is :   . . .   going to be installed   E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution). Collapsible fieldsets error for previous dates List of chat rooms with host stackoverflow.com is not the same as the actual list of rooms on stackoverflow.com chat itself Generally chat.&lt;site&gt;.com redirects to chat.stackexchange.com/?tab=site&amp;host=&lt;site&gt;.com and go for same thing but here you can check your self:       Both are Different!  If one visits  and select site: StackOverflow, then it outputs  and they miss rooms that are only listed by .  So, why does this happen? Is this a bug?  Note: I found this while  for general chat-room at StackOverflow; see chat-discussion How do I change my username? I'm writing my thesis using MiKTeX and WinEdt 9 for Windows. I write the different chapters individually, and then I have a "masterfille" with the documentclass "report" where I use the command "input" to load the individual chapters. In other words there is only a preamble in the "masterfile".  I have a very general problem which is the following: No matter what kind of mistake I do whatsoever, I will first receive the correct error message, say I forgot to end a table or something (it really does not matter). If I correct this mistake and try to recompile without using the option "Erase Output Files" in Winedt, I will always and very quickly receive a new error message, often containing a mysterious "Undefined control sequence". Having to do this every time and to recompile plus run the Biber (L+B+L+L+D) is very time consuming, since the document is very large. Any ideas what the problem could be? Here is the preamble of my "masterfile" (I am probably including irrelevant information, sorry for that):   \documentclass[12]{report} \setcounter{tocdepth}{2} \usepackage{layout} \setlength{\voffset}{-0.75in} \setlength{\headsep}{5pt} \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{csquotes} \usepackage{authblk} \usepackage [T1]{fontenc} \usepackage[backend=biber, style=authoryear-comp]{biblatex} \usepackage{xyling} \usepackage[super]{nth} \usepackage{xspace} \usepackage{cgloss4e} \usepackage{abstract} \usepackage{array} \usepackage{multirow} \usepackage{hyperref} \usepackage{xr} \usepackage{gb4e} \noautomath  \let\eachwordone\itshape   \makeatletter \pretocmd{\@footnotetext}{     \@noftnotefalse\setcounter{fnx}{0}%     \renewcommand{\thexnumi}{\roman{xnumi}}     }{}{} \apptocmd{\@footnotetext}{     \@noftnotetrue     \renewcommand{\thexnumi}{\arabic{xnumi}}     }{}{} \@ifpackageloaded{bidi}{% \pretocmd{\@LTRfootnotetext}{     \@noftnotefalse\setcounter{fnx}{0}%     \renewcommand{\thexnumi}{\roman{xnumi}}     }{}{} \apptocmd{\@LTRfootnotetext}{     \@noftnotetrue     \renewcommand{\thexnumi}{\arabic{xnumi}}     }{}{} } \makeatother     \addbibresource{ref.bib}    \title{The origins of Old Romance V2: A comparative study of Late Latin, Old French and Middle High German} \author{XXX} \date{\today}   \begin{document}  \tableofcontents   \chapter{Old French} \input{chapters/OldFrench2}  \printbibliography  \end{document} Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Suppose we have functions $f:A→B$ and $g:B→C$. Prove that if $f$ and $g$ are invertible, then so is $g \circ f$. Is the converse true? I.e., if $g \circ f$ is invertible, does it follow that $f$ and $g$ are both invertible?  I know that if $f$ and $g$ are invertible, then they are both injective and surjective. Therefore, I assume from this that I have to show $g \circ f$ is injective and surjective as well, thus proving it is invertible. However, I don't know how to show $g \circ f$ is bijective based only on the information I have. Could Hermione really free the Hogwarts house-elves with her knitted hats? Say if $A$ is an $n \times  n$ matrix, why is it that if $A^{T}A$ is positive definite, the matrix $A$ is then invertible? All I know is $A^{T}A$ gives a symmetric matrix but what does $A^{T}A$ is positive definite tell or imply or hint about the matrix $A$ itself that leads to the fact that it will be invertible?