Prove by induction that $2^n\le (n+1)!$ for all $n\in\Bbb N$.  What I've done so far is prove for $n=1$: $$2^1 \le (1+1)!,$$ $$2 \le 2,$$ which is correct Then I tried to prove for $n+1$, in other words, I want to get here: $$2^{n+1}\le(n+2)!.$$ So, I multiplied everything with $2$: $$2\cdot 2^n \le2\cdot (n+1)!,$$ $$2^{n+1} \le2(n+1)n!.$$ So I already have what I wanted in the left part of the inequality, but I'm stuck for the right part. Can someone help me? Thanks!
I need help with this exercise. What I've done so far is prove the exercise when $n=1$. So:  $$n=1$$ $$1!\ge2^{(1-1)}$$ $$1\ge2^0$$ $$1\ge1$$ Which is true  Therefore, now that I assume that the assumption is correct, I want to prove that with $n+1$, it will also be true. So, what I've done now is:  $$P(n) \implies P(n+1) $$ $$n!+n+1\ge 2^{(n-1)}...$$ And my  problem is that I do not know how to add the $n+1$ in the right side of the equation, therefore I'm not been able to finish the exercise.  Thank you so much for your help. If something's not very clear, please let me know. I'll try to be clearer next time :)
I've got a directory shared on my network, using Samba. People on the netwerk use this Samba share to download files from my share to their computers. I'm interested in what files are downloaded most, and by which users they are downloaded. I'd like to see this in real time, if possible. Is there a program that allows me to do this? If yes, how can I use it?    I'm using Ubuntu 11.10. Suppose that I have a &lt;div&gt; that I wish to center in the browser's display (viewport). To do so, I need to calculate the width and height of the &lt;div&gt; element.   What should I use? Please include information on browser compatibility. I just rememebered a case when I was still in school: we had to write an essay about a specific topic and one guy put big parts of his essay on Wikipedia before handing it in. When the teacher checked for plagiarism she indeed found big parts of his text in the Wikipedia article and thus accused him of plagiarism.  How would that situation be with journal papers (or other "official" ways of publishing)?  Can I reuse part(s) of texts I wrote myself but that are available publicly/online like on Wikipedia, my Blog, university homepage, etc.?  (Maybe consider that pseudonyms are used on Wikipedia, Blogs, etc.) What is "root" and how can I become it? When I ran a command, I got the following error:     You need to be root to perform this command.   What is root and how can I become it? A friend of mine misplaced his passport while staying in Jeju island. Normally I would advise him to just travel to Seoul, get an emergency passport from his embassy and fly back home without issues, but I know that Jeju island has a special immigration regime.   Are there any immigration checks on the way from Jeju island to Seoul? If not, can one travel that route without a passport? Is there a way to show Spotlight indexing status/progress in Yosemite? In previous versions of OS X, you could click the Spotlight icon in the menu bar to show the status of the Spotlight indexing, and it would show an estimate ("6 hours left"), along with a progress bar.  I've updated to Yosemite yesterday, and I noticed that Spotlight is not showing some expected results, i.e. files I know to be there, but it just doesn't find them if I type in their name and file type ("kind:excel vacation").  When I start typing in the Yosemite Spotlight window, it shows a small progress bar under the search box, but there's no estimate.    Is there another way to show the progress or an estimation on how long the indexing is going to take? How to delete a non-empty directory in Terminal? How do I delete the following directory?  I typed:  rmdir lampp   This error comes up:  rmdir: failed to remove `lampp': Directory not empty   Is there a command to delete all the files in the directory and delete the directory folder? Would the latter be an issue with the funding body, when the project proposal and budget only covers the expenses for hiring one postdoc? Note: it is the supervisors who got the grants, not the postdocs themselves. Radius of Convergence of a given Power series Suppose I have a 6 GB page file on an SSD and another 6 GB page file on a traditional mechanical 5400rpm SATA drive.  Does Windows take into account the difference in the performance characteristics between the two drives when deciding which page file to use?  If it matters, I'm asking about Windows 7 and higher.  I reviewed this  but it doesn't appear to factor in the case of dissimilar storage speeds. How do I get started playing Dungeons and Dragons 4th edition by myself? I've been wanting to start playing D&amp;D for some time, and I was wondering how I could get started by myself with relative ease. I own the first two player handbooks, and a set of dice. JavaScriptSerializer - JSON serialization of enum as string How to change boot priority? I have a laptop with Ubuntu installed as the only OS. I also have a USB stick with a Windows 7 installation on it. I want to install Windows from the USB stick on my laptop but when I try to restart the laptop to get access to BIOS, I can't seem to find the command for it, its like it skips everything and just says Ubuntu and then takes me to the login screen.  How do I change the boot priority to USB in Ubuntu? We were taught that an independent clause must have a complete thought/idea.     This is the only mammal that can fly.     How can "This is the only mammal" be treated as an independent clause?            It was a grammar lesson and there was no context there. In this situation how can I decide that it is an independent clause? Though for sure, there should be an independent clause in a correct sentence, and the clause starting with 'that' is not independent. But by this way, I can't fit it with the rule that every independent clause must have a complete thought. A simple sentence must have an independent clause. If the sentence has no independent clause how we can call it a sentence? Can even two dependent clauses form a sentence? Why does `last` show '{' and '|' in the TTY field? Is there a way to load every label data and every SObject description data using only Javascript without any Apex Server-side code?  How do I import * from @salesforce/schema or @salesforce/label?  I am looking for a way to get all labels or all SObject describe information details to retrieve in LWC at once.  I can see that such syntax is provided by ES6 and is used widely by Salesforce itself, however, it is not possible to use the following syntax in LWC:   import * as schema from '@salesforce/schema'   import * as label from '@salesforce/label'   or this syntax  import {label, schema} from '@salesforce';   since both versions generate error     LWC1512: Missing resource value for @salesforce"   or      LWC1512: Missing resource value for @salesforce/label   Is there a way to load every label data and every SObject description data using only Javascript without any Apex Server-side code?  By the way, if we look as Salesforce Playground we could notice that   the same syntax is used to import everything from lwc  import * as Engine from 'lwc';   however, if we try to insert such line into any of Lightning Web Component, we would receive the following error     LWC1001: Invalid import. Namespace imports are not allowed on "lwc",   instead use named imports "import { LightningElement } from 'lwc'"   or   LWC1517: Invalid LWC import The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. When running Internet Explorer from the the modern UI, I can't find out how to access tabs.  I can create tabs using Ctrl + T, but I'm not sure how to switch between tabs, see what tabs I have open, etc.  How do I navigate between tabs when using the modern UI Internet Explorer? Can SMT give worse performance than normal CPU? Change style properties of several layers simultaneously?