Prove the sequence $a_n=(1+\frac{1}{n})^n$ is monotonically increasing and bounded from above. USING INDUCTION
Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
semi filled circle Keep items in inventory on death in Minecraft I once played a vanilla Minecraft adventure map (called "Herobrine's Mansion") that prevented the player from losing their items on death. Which commands can be used to prevent the player from losing their items on death (in vanilla Minecraft)? Fisher Information of a function of a parameter bash variables in for loop range I want to print list of numbers from 1 to 100 and I use a for loop like the following:  number=100 for num in {1..$number} do   echo $num done   When I execute the command it only prints {1..100} and not the list of number from 1 to 100. I have to prove that a function $f:]0,1] \rightarrow \Bbb R$ : $$ f(x) = \begin{cases} \frac1q,  &amp; \text{if $x \in \Bbb Q$  with $ x=\frac{p}q$ for $p,q \in \Bbb N$ coprime} \\ 0, &amp; \text{if $x \notin \Bbb Q $} \end{cases} $$  is discontinuous in every point $x \in \ ]0,1] \cap\Bbb Q$.  And then to consider $x \in \ ]0,1] \backslash \Bbb Q$ and prove that it is continuous.  For now I learned different ways to prove continuity (epsilon-delta, sequences), but I'm never sure what would be better to use in each different case.  I wanted to prove the discontinuity by using sequences: $$\forall x_n \quad x_n\rightarrow a \quad \Rightarrow \quad f(x_n) \rightarrow f(a)$$  I tried creating a sequence $ x_n=\frac1n + a$, we know it converges to $a$ but $f(x_n)\ $ doesn't  converges to $\ f(a)$ because there would still be some points not in our set (irrational numbers that creates gaps).  But I don't think it works, so I'm asking you if you could help me solving the two questions. I have the following pieces of hardware I would like to network mostly for file sharing (media primarily): 1 gaming laptop, 1 i7 (2.3Tb storage) desktop, 1 P4 desktop (500g storage), and 1 250gig PS3.  I am looking to setup a linux host (with a PS3 media server) with the i7 desktop but am concerned about leaving it on all the time. What stripped down, low energy usage distro can I couple with which windows &amp; PS3 friendly software?  Also, how should i reformat my i7 (in terms of partitions, RAID options, whatever)? It has 2 1Tb drives (likely adding a third), and 1 320 gig. When we want to open an application or file from the terminal, we type, say,  okular file.dvi   This opens the application, but also shows the status of the application. We cannot close the terminal, because it kills the process. Unfortunately, if you're trying to, for example, create a LaTeX file, you will need one tab for the text editor, one for the dvi file, and so on. And if you're trying to open all windows from the terminal, you can forget it. I'm trying use the terminal as much as possible, and while I have Yakuake, it is still a bother having so many tabs and seeing which of those have an application I've killed and so on.  So, is there a way to open an application/file from the terminal so that the status doesn't show and immediately gives the prompt so that we can use it to open more applications? Circle Rolling on Ellipse I've gotten interested in describing a circle rolling on an ellipse; specifically, the curve traced out by a point on the circumference of the circle. I want a symbolic solution to the general case, radius $r$, axes $a$ and $b$. I've written nine polynomial equations in terms of various angles and lengths.  Exactly what "solution" means is subject to debate.  Let $(u,v)$ be the point on the circle. Similar to the cycloid, I would like an equation for $u$ in terms of a "natural" angle in the problem.  Similarly, an equation for $v$.  Perhaps it is necessary to have a differential equation, so maybe $du/dt$, $u$, and $t$, where $t$ is an angle in the problem.  I would have thought this was known, but I can't find it anywhere. How to delete GRUB entirely from GRUB rescue and boot Windows only When transiting between two flights, will my luggage be automatically transferred? I just read a book "Linear Algebra and Its Applications 4E" by David C. Lay. The book is (should be) LaTeX typeset. The matrix and equations are so nice, please see the following 3 examples:        It's so beautiful, so I want get the same effect inside my paper. Here is my code:   \usepackage{amsmath} \usepackage{colortbl} \newcommand\y{\cellcolor{clight2}} \definecolor{clight2}{RGB}{212, 237, 244}%  %%% \[A=\left[ \begin{array}{ccccc} \y 0 &amp; -3 &amp; -6 &amp; 4 &amp; 9 \\    -1 &amp; \y -2 &amp; -1 &amp; 3 &amp; 1 \\ -2 &amp; -3 &amp; 0 &amp; \y 3 &amp; -1 \\ 1 &amp; 4 &amp; 5 &amp; -9 &amp; -7      \end{array} \right]\]  \[A=\left[\begin{array}{cccccc} \rowcolor{clight2} 3 &amp; -9 &amp; 12 &amp; -9 &amp; 6 &amp; 15\\ 0 &amp; 2 &amp; -4 &amp; 4 &amp; 2 &amp; -6 \\ 0 &amp; 3 &amp; -6 &amp; 6 &amp; 4 &amp; -5 \end{array} \right]\]   And the result:     But how to add these arrows?? How to pad zeroes to a string? What is a Pythonic way to pad a numeric string with zeroes to the left, i.e. so the numeric string has a specific length? How do I remove duplicates from a C# array? I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? I was just wondering whether the following statement is correct.  Let R be a ring and M a noetherian R module. Then M is finitely generated. Too many requests? I went to stackexchange.com and clicked on SO.  After a few minutes I got a message saying "We have detected too many requests from this IP.  If this is an error contact us."  That was the second SE page I tried to visit.  Second only to SE.com  I moved to another machine and it's working now.  The first machine usually works, I don't know if a got assigned a blacklisted IP or what.  Anyone know what's going on?  EDIT: I would really rather not post my IP on the web, but if a diamond or dev asks I'll email it.  EDIT 2: I looked at badp's dupe suggestion and can verify that I have no scripts frequently calling SO. Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$,  where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\sigma^2$ and the sequences $\{X_i\}$ and $\{Y_i\}$ are independent.  Then, for all real numbers $x$, show that, as $n\to \infty$,  $$\mathbb{P}(U_n\leq x) \to  \int_{-\infty}^{x} \frac{1}{(\sqrt(2\pi))} \:e^{-\frac 12{u^2}}\mathrm du $$ Approach  I tried to carry out the integration on the right, but this does not integrate nicely. It ends up involving an error function, so I figured this was the wrong was to go.  Should I be looking at convergence in probability to answer this question? Is Chebyshev's inequality relevant here? If so, how and how does that relate to the integral? Update on Ubuntu - coloured lines around everything Everyday I check for updates on Ubuntu 16.04 LTS. I updated to 4.4.0-65-generic today and now on everything I'm getting thick rainbow coloured lines around every dialog box in every program. My setup is pretty standard. Skylake i5-6500, Nvidia 950gtx. Does anyone know how to fix it? How to import a .sid Raster File in QGIS 2.6.0 On Linux Mint 17 I need some help importing MrSID Raster files in QGIS. I am running 2.6.0 on Linux Mint 17. After an internet search I have found a couple of resources that are perhaps outdated. The most promising is found here :    However when I attempted to follow the instructions my terminal and I could not find the file "libgdal-mrsid-src" on either the stable or unstable Ubuntu GIS repos. Has anyone had success or could lead me in the right direction? I have a valid B1/B2 visa valid to US till March 2019 on Indian passport, I changed my citizenship to Australian. Is my B1/B2 visa on Indian passport valid? Is it physically possible to build a long zoom (17-300 mm for example) at an aperture value of around 1.4 or 1.2? I'm curious if the physics allow a company to build such a lens of course with good IQ, minimal color aberration, maximum sharpness, and little to none lens vignetting or edge softness