Proof.  Base case. When $n = 6$, LHS = $6! = 720$ and RHS = $6^{3} =  216$. Thus LHS > RHS.  Inductive step. Assume that $m! &gt; m^{3}$ for some fixed integer $m$ such that $m \ge 6$. We must show that it also holds true for $(m+1)$, i.e, $(m+1)! &gt; (m+1)^{3}$. Observe that   \begin{align*} (m+1)! &amp;= (m+1) \cdot m! \\ &amp;&gt; (m+1) \cdot m^{3} \text{$\quad$(by the inductive hypothesis)}\\ \end{align*}  (still unfinished)    I know it boils down to showing that $(m+1) \cdot m^{3} &gt; (m+1)^{3}$ but I'm unsure on how to proceed.
I'm given a inequality as such: $n! &gt; n^3$ Where n > 5,   I've done this so far:  BC: n = 6, 6! > 720 (Works)  IH: let n = k, we have that: $k! &gt; k^3$  IS: try n = k+1, (I'm told to only work from one side)  So I have (k+1)!, but I'm not sure where to go from here.  I've been told that writing out: $(k+1)! &gt; (k+1)^3$ is a fallacy, because I can't sub in k+1 into both sides, but rather prove from one side only.    Any Help on how to continue from here, would be much appreciated.
whether they are complete metric space How can I find the right lens cap for my lens? I own a Nikon D3000 Camera with the AF-S 18-55mm VR Kit Lens. Last week I lost the lens cap and now I want to buy a new one.  But when I went to , I got confused with the different ranges of lens names —  LC-62, LC-78, etc.  My concern is what if I order one lens cap and it won't fit correctly on my lens?  If you guys have an D3000, can you please let me know how to find the correct lens cap names? Accessing inherited variable from templated parent class Consider the following code:  template&lt;class T&gt; class Foo { public:   Foo() { a = 1; }  protected:   int a; };  template&lt;class T&gt; class Bar : public Foo&lt;T&gt; { public:   Bar() { b = 4; };    int Perna(int u);  protected:   int b;  };  template&lt;class T&gt; int Bar&lt;T&gt;::Perna(int u)  {    int c = Foo&lt;T&gt;::a * 4; // This works   return (a + b) * u;    // This doesn't }   g++ 3.4.6, 4.3.2 and 4.1.2 give error   test.cpp: In member function `int Bar&lt;T&gt;::Perna(int)': test.cpp:25: error: `a' was not declared in this scope   g++ 2.96 and MSVC 6, 7, 7.1, 8 and 9 accept it, as do (at least) older Intel and SGI c++ compilers.   Do the new Gnu C++ compiler obey the standard or not? If they do, what is the rationale behind that inheriting class is not able to see a protected inherited member variable?  Also, if there's  int A() { return a; }    in Foo, I get error  test.cpp:25: error: there are no arguments to A that depend on a template parameter, so a declaration of A must be available test.cpp:25: error: (if you use -fpermissiveâ, G++ will accept your code, but allowing the use of an undeclared name is deprecated)   when I try to use it in a member function of Bar. I find that curious as well: Bar inherits Foo, so I think it's obvious that A() in scope of Bar is Foo::A(). showing $\psi: R\to \mathbb C$ is ring isomorphism. onKeyUp not firing for ESC in IE How many ways to arrange people on a bench so that no woman sits next to another woman? Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain. In calculating work done by a constant force over a constant distance, why doesn't the subject's initial velocity matter? Assume a point-mass $m$ is travelling in a straight line, and a force $F$ will act on $m$ (in the same direction as $m$'s velocity) over a constant distance $d$; why doesn't $m$'s velocity matter to the calculation of work done on $m$ by $F$? Work is defined such that, in this example, the work done by $F$ on $m$ is equal to $Fd$, but it seems that if $m$ were moving slower, it would spend more time in the field, allowing $F$ more time to act on $m$, thereby doing more work. In fact, if $m$'s velocity were very great, it would hardly spend time in $F$'s field at all (so very little work done). Maybe I misunderstand work; can someone address this confusion of mine? How to add desktop shortcut for URL, ignoring web app manifest? Goal:  Have a desktop shortcut that opens  , ignoring the contained web app manifest that otherwise "overrides" the saved shortcut to  and with "display": "standalone".  I had this working previously on an older Android phone before a replacement / upgrade.  Prior to the upgrade, I was able to long-press and open individual articles in new tabs - as well as copy link shortcuts to specific articles, as well as all of the other features and functionality that Chrome has to offer.  Here is just a partial selection of references of other users complaining about the same:       Using  provides a temporary successful work-around.  However, I can't find a successful way to add a link for this to my home screen.  The "Add to Home screen" option within Chrome instead saves a link to  - and in such a way that it opens as a web application instead of a web page within Chrome:       On a desktop operating system, this would be a non-issue.  (Create shortcut, enter desired URL, done.)  Attempt 1: Provide my own Manifest  So I created my own manifest - editing  to change the start_url attribute to https://news.google.com/news/i#0, and the display attribute from standalone to browser.  I then created my own HTML page that simply contained the below - then self-hosted both on a local web server:  &lt;html&gt;         &lt;head&gt;                 &lt;Title&gt;Google News&lt;/Title&gt;                 &lt;link rel="manifest" href="/news/manifest.json"/&gt;         &lt;/head&gt;         &lt;body&gt;                 Test         &lt;/body&gt; &lt;/html&gt;   However, this doesn't work:  Despite the shortcut being saved with the desired name and icon, it links back to my self-hosted HTML page, and not the indicated start_url.  Unfortunately, it appears that cross-domain URLs are not allowed per the manifest intent - as per  and  .  Attempt 1b: MITM website  Within the confines of my own local network, I figured I could try to  (MITM) the website to temporarily provide a desired manfiest to use for addition to the home screen.  Unfortunately, the website uses  here, which appropriately blocks the attempt.  Regardless - this is way too much work for what should be a simple effort.  Attempt 2: Widget Bookmark  Unfortunately, it looks like the 1x1 bookmark widget was removed in Chrome 48, as per  .  The suggested work-around is to use the "Add to Home screen" feature within Chrome.  Goto 10 / see beginning.  Attempt 3: Use the default-provided "Internet" browser  Unfortunately, using "Add shortcut on Home screen" in the default-provided "Internet" browser produces the same result as Chrome.  Consideration 4: Use a custom app / widget  Could probably write and install a custom application or widget that could be added to the home screen to meet the goal.  Or don't re-invent the wheel, and use an existing one.  Some possibilities:   "More Shortcuts":  "Shortcut Customizer":    Conclusion  How can someone create a desktop shortcut to an arbitrary URL, ignoring a provided web app manifest? Is there anything in RAW that supports spell level as a character concept, or alternatively that supports it as solely a player concept?  On the one hand, I see nothing that explicitly states that a character knows that there are spell levels. On the other hand, I don't see how a spellcaster could manage spell slots without knowing about spell levels. Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\ $ [Double Factor Theorem] Downloading PlayOnLinux I'm downloading PlayOnLinux to run Paint Tool SAI, but I cannot open PlayOnLinux. When I click on it to open it, a dialog box comes up that says: "PlayOnLinux cannot find curl (from cURL) You need to install it to continue". How do I install this?   I'm referring to this article: How does electricity propagate in a conductor? Why is Python 2.7 still the default Python version in Ubuntu? According to official documentation of Python, Python2.7 support will end near in future.     DEPRECATION: Python 2.7 will reach the end of its life on January 1st, 2020. Please upgrade your Python as Python 2.7 won't be maintained after that date. A future version of pip will drop support for Python 2.7. More details about Python 2 support in pip, can be found at    Why is Ubuntu not using Python3 as default in their upcoming version?  $ cat /etc/os-release  NAME="Ubuntu" VERSION="19.04 (Disco Dingo)" ID=ubuntu ID_LIKE=debian PRETTY_NAME="Ubuntu 19.04" VERSION_ID="19.04" HOME_URL="https://www.ubuntu.com/" SUPPORT_URL="https://help.ubuntu.com/" BUG_REPORT_URL="https://bugs.launchpad.net/ubuntu/" PRIVACY_POLICY_URL="https://www.ubuntu.com/legal/terms-and-policies/privacy-policy" VERSION_CODENAME=disco UBUNTU_CODENAME=disco     $ python Python 2.7.16 (default, Apr  6 2019, 01:42:57)  [GCC 8.3.0] on linux2 Type "help", "copyright", "credits" or "license" for more information. I find I always have trouble scaling the amount of time the eggs should be boiled, cooled, etc. for different-sized batches of hard-boiled eggs.  I've heard a variety of "folk lore"-type rules for how it should be done, but what's really the right way? I wanna create a new partition from the existing root partition.My root partition is about 250GB.I wanna create a new partition say 150 GB taking space from this root partition.Can anyone help me with the procedure on doing this?  output: fdisk -l  karthick@karthick:~$ sudo fdisk -l [sudo] password for karthick:   Disk /dev/sda: 250.0 GB, 250000000000 bytes 255 heads, 63 sectors/track, 30394 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x28219744  Device Boot      Start         End      Blocks   Id  System /dev/sda1   *           1          61      487424   83  Linux Partition 1 does not end on cylinder boundary. /dev/sda2              61       30395   243650561    5  Extended /dev/sda5              61         124      498688   82  Linux swap / Solaris /dev/sda6             124         186      498688   82  Linux swap / Solaris /dev/sda7             186       30395   242651136   83  Linux Real Analysis, Existence of a Limit Boundedly:$|\int_0^b\frac{sin(ax)}{x}dx | \le c$ I need to show that there exists a $c$ such that $|\int_0^b\frac{sin(ax)}{x}dx | \le c$, where $a$ is any real number, and $b$ is any real number greater than 0. Also I need to show that the smallest $c$ possible is $c=\int_0^\pi\frac{sinx}{x}dx$. I know that $\int_0^\infty \frac{sin(ax)}{x}dx = sig(a)\frac{\pi}{2}$ where $sig(a)$ literally just means positive one or negative one, depending on the sign of $a$. I think the fourier tranform of the characteristic function on the interval $(0,a)$ is $\frac{sin(ax)}{x}$. After that, I am unsure about how to proceed. Thanks for your help! What is a decoupling capacitor and how do I know if I need one? What is a decoupling capacitor (or smoothing capacitor as referred to in the link below)?  How do I know if I need one and if so, what size and where it needs to go?    mentions many chips needing one between VCC and GND; how do I know if a specific chip is one?   Would an  4-bit parallel access shift register used with an Arduino need one? (To use my current project as an example) Why or why not?  I feel like I'm starting to understand the basics of resistors and some places they're used, what values should be used in said places, etc, and I'd like to understand capacitors at the basic level as well. How do I get breading to stick to pre-cooked chicken for pan frying? Java Array, Finding Duplicates