Prove $\sqrt{n+1} &lt; \sqrt{n} + 1 $ for all $n \ge 1$.  I have proven the base step for $n = 1$. $\sqrt{2}$ is less than $2$. The inductive hypothesis is $\sqrt{n+1} &lt; \sqrt{n} + 1$. From here, I am unsure of how to proceed. How can I show that the inequality $\sqrt{(k+1)+ 1} &lt; \sqrt{k+1} + 1$? After simplifying the expression $\sqrt{(k+1)+1}$, it becomes $\sqrt{k+2}$. Perhaps I can use this somehow to come up with an expression? The right hand side does not have a radical over the 1, so I do not think it is a duplicate question that has an answer in the suggested question.
I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this?  $$\sqrt{a+b}  \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$  Moreover, what happens with more variables? Say:  $$\sqrt{x_1+x_2+...+x_n} \le \sqrt x_1 +\sqrt x_2 + ... + \sqrt x_n $$  with $ x_i \in\mathbb R_+ \forall i \in \{1,2,...,n\} $  Or when $i = \mathbb N$  PS: I tried looking on the internet for this but I don't know how I would call this.
How to hide Authoring information,Revision information,Publishing options tabs in create/edit node for any content type I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot Why does the division algorithm work for converting between number bases? I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks How to install Telegram Where can I find medicine? I know it is possible to be stung by bees in Animal Crossing: New Leaf.   I learned that the hard way when I was trying to find my furniture tree. Now, I have an eye swollen shut and all of my towns people are making fun of me. They're all being kind enough to inform me that I need Medicine, but they seem to be enjoying themselves at my expense too much to tell me where to actually find the Medicine.  Where can i get medicine? I'd really like it if my townspeople stopped making fun of me. I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0&lt;a&lt;1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz=\left(\int_{L_{1}}+\int_{L_{2}}+\int_{C_{R}}+\int_{C_{\epsilon}}\right)\frac{\ln z}{z^{a}(z+1)}dz$$ The function has a pole at $z=-1$. Then, by residue theorem, the whole integral becomes  \begin{align*} \int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz&amp;=2\pi i\text{res}(\frac{\ln z}{z^{a}(z+1)};-1)\\&amp;=2\pi i\lim_{z\rightarrow-1}(z+1)\frac{\ln z}{z^{a}(z+1)}\\&amp;=2\pi i\lim_{z\rightarrow-1}\frac{\ln z}{z^{a}}\\&amp;=2\pi i\frac{\ln e^{i\pi}}{e^{i\pi a}}\\&amp;=2\pi i\frac{\ln1+i\pi}{e^{i\pi a}}\\&amp;=-\frac{2\pi^{2}}{(e^{i\pi})^{a}} \end{align*} On $L_1$, $z=xe^{i\epsilon}$ where $\epsilon\leq x\leq R$  On $L_{2}$, $z=xe^{i(2\pi-\epsilon)}$ where $\epsilon\leq x\leq R$  On $C_{R}$, $z=Re^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  Lastly on On $C_{\epsilon}$, $z=\epsilon e^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  So I tried to compute them separately and I anticipate the curves will both go to $0$ by ML inequality, and two linear integrals will consist of the term $\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx$ and hence simply solve for it. But I'm having trouble physically solving it out. I think the log is throwing me off. Can anybody please help me? I am talking about when you click on the icons in the dock This command gsettings set org.compiz.unityshell:/org/compiz/profiles/unity/plugins/unityshell/ launcher-minimize-window true  which works for Unity Launcher, (obviously) does not work.  Related to: New passport and entrance Schengen I am from Brazil and travel a lot in Schengen countries. Under the rules of Schengen visa (3 months stay in a 6 months period). In my passport are the entrance and exit stamps, also to chec the days I stay in Schengen.  I need a new passport and will get this in Brazil. So in this new passport are NOT the stamps. Does this mean I start new with the Schengen visa rule? I seems like. The new passport is empty. Or do I miss something? It would be great because I am almost on my 90 days stay in the period of 180 days. We all know how important proper attribution of ideas is. At the same time, certain things have become basic enough that citing the paper where they were first discussed is overkill: to give an extreme example, if you need to do some differentiation in your math/physics paper, you don't need to go and cite Newton and Leibniz. Now, on occasion students ask me how one can determine if a piece of knowledge is common enough that they can forgo a citation. The rule of thumb I give them is:     If it is something that is explained in a standard first-year undergrad textbook, then anybody who is going to read your papers knows about it and you don't need to provide a citation.   [Here I want to emphasize that I give this to them as a rule of thumb, and I always tell them to ignore it and provide the relevant citation if they think it is necessary to do so in a specific case]  Are there better or alternative ways of drawing the line? How is the amplitude of light related to number of photons? If you increase the number of photons leaving a light source you increase the brightness/intensity, so it could be said that intensity is directly proportional to number of photons. However intensity is directly proportional to amplitude squared, so this leads me to wonder that if you increase the amplitude of a light wave, are you just creating more photons rather than affecting each individual photon in some way? I keep seeing places refer to the "multiverse" repository as a place I can get software, how can I enable this repository? Please specify how to do this graphically and via command line. I am under DDoS. What can I do? This is a  about DoS and DDoS mitigation.   I found a massive traffic spike on a website that I host today; I am getting thousands of connections a second and I see I'm using all 100Mbps of my available bandwidth.  Nobody can access my site because all the requests time out, and I can't even log into the server because SSH times out too!  This has happened a couple times before, and each time it's lasted a couple hours and gone away on its own.  Occasionally, my website has another distinct but related problem: my server's load average (which is usually around .25) rockets up to 20 or more and nobody can access my site just the same as the other case.  It also goes away after a few hours.  Restarting my server doesn't help; what can I do to make my site accessible again, and what is happening?  Relatedly, I found once that for a day or two, every time I started my service, it got a connection from a particular IP address and then crashed.  As soon as I started it up again, this happened again and it crashed again.  How is that similar, and what can I do about it? Clicking links in pdfs opened with Okular opens Abiword (instead of the default browser) I work with Xubuntu 12.04 and use Okular (version 0.14.3) to view pdf files. If I click a web link in a pdf file (created with "pdflatex" using the "hyperref" package), "Abiword" (It's a word-processing software, like Libreoffice writer) is opened instead of my default browser google-chrome. How can I change this behavior? The settings in Okular do not seem to provide a solution.   (Note that I set google-chrome as the preferred application for web browsing under Settings -> Preferred Applications, so that's not the problem). Marketing Cloud contact builder shows 9 times more contacts How to secure database passwords in PHP? When a PHP application makes a database connection it of course generally needs to pass a login and password. If I'm using a single, minimum-permission login for my application, then the PHP needs to know that login and password somewhere. What is the best way to secure that password? It seems like just writing it in the PHP code isn't a good idea. $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$ 80's Animated movie: An old man joins mission to save a boy stranded on a hostile world How to render world position pass A world position pass is a pass used for composting other elements into the scene after the initial render. In a world position pass the R, G, B values correspond to the X, Y, Z axis (or X, Z, Y). A world position pass could look something like this. (Image form another 3d app)    Is there a way to render out corresponding world space coordinates of each pixel? Custom Button on Lightning Experience I created a custom button Send Email in Lightning Experience. The problem is that button in visible in feed section layout, but I want to put it in the upper right corner of page layouts near Edit, Clone, Delete ecc....  How Can I do that? Thanks  I add an image that explain the situation.    EDIT: I would like to explain you better my problem.  I have created three custom quick action:   I have added them to the case page Layout as you can see    I would like to understand the reason why I don’t see these three custom action in the Page Layout in the section in the top right Why do I get "Exception; must be caught or declared to be thrown" when I try to compile my Java code?