I kinda understand the logic and motivation behind the proof, but what bothers me is the fact, why is the base case (the first statement that we write) is always $P(1)$ when we are proving a proposition is true for all $\mathbb{N} $? Why can't it be anything other than $1$? What about $P(5)$ or something? Does it change anything?
I attempted a proof of mathematical induction using an arbitrary base case, but was unsuccessful (and hence this question). Below is what I was trying to do and along with my thinking; if anyone can point me in the right direction I'd appreciate it.  The induction I am using: Let $P$ be a property about the natural numbers $\mathbb{N}$ with $0\in\mathbb{N}$, and let $P(n)$ denote the statement that the property $P$ holds for $n\in\mathbb{N}$. Suppose $P(0)$. Furthermore suppose that for each natural number $k$, $P(k)$ implies $P(k+1)$. Then $\forall nP(n)$.  What I am trying to prove: Let $P$ be a property about the natural numbers $\mathbb{N}$ with $0\in\mathbb{N}$, and let $P(n)$ denote the statement that the property $P$ holds for $n\in\mathbb{N}$. Suppose for $n_0\in\mathbb{N}$, $P(n_0)$. Furthermore suppose that for each natural number $k\geq n_0$, $P(k)$ implies $P(k+1)$. Then $\forall n\geq n_0 P(n)$.  My attempted proof: define $Q(n)$ to be $n\geq n_0 \to P(n)$. Then we wish to prove $\forall nQ(n)$. We induct on $n$.  Base case (for ordinary induction): $Q(0)$ is $0\geq n_0\to P(0)$. Since $n_0\in\mathbb{N}$, $0\geq n_0$ implies that $n_0=0$. Since $P(n_0)$, $P(0)$, which proves the base case.  Inductive step: we want to show $\forall n(Q(n)\to Q(n+1))$. To do this, we assume $k\geq n_0 \to P(k)$ and try to show $k+1\geq n_0 \to P(k+1)$.  Since $k\geq n_0 \to P(k)$, we first prove the case where $k\geq n_0$ and $P(k)$. By the hypothesis of the proof, we see that $P(k+1)$, which proves the case for $k+1$.  This is where I am having trouble: For $k&lt;n_0$, I am unable to show the implication for $k+1$.  So my questions would be: (1) is the overall approach for the proof correct? (2) If so, how might I go on to prove the case when $k&lt;n_0$?  Thanks in advance. (This is not homework, by the way.)
How to trigger workflow from button in O365 I created a script editor webpart. I added a button which start a js function in the onClick event. I would like to start a workflow which is implemented on some list. Is this possible? StringPattern to remove markers around text What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? It can be proved that there are two irrational numbers $a$ and $b$ such that $a^b$ is rational (see ) and that for each irrational number $c$ there exists another irrational number $d$ such that $c^d$ is rational (see ).  My question is: Is there an irrational number $a$ such that $a^a$ is rational (and how could you prove that)? Related:   I know I must describe my past and current research in my personal/research statement or statement of purpose. But do I use technical terms?  I was given the advice that since the people reading my statement are mathematicians (I am applying to US for applied mathematics), they should know what I am talking about.  Although I am applying to applied mathematics, it would also be nice to hear from other fields (biology, computer science, engineering etc) about this. Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino How to vertically center a "div" element for all browsers using CSS? How can I write the name show in axis of 3D array using TiKz I am drawing a 3D array like the attached picture, which actually describes a multidimensional array. However, I can not put the name of the axis and take the value from the matrix like the attached picture.  This is the code that I have:  \documentclass[margin=5mm]{standalone} \usepackage{tikz} \usetikzlibrary{matrix,calc}  \begin{document} \begin{tikzpicture}[every node/.style={anchor=north east,fill=white,minimum width=1.4cm,minimum height=7mm}] \matrix (mA) [draw,matrix of math nodes] { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mB) [draw,matrix of math nodes] at ($(mA.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mC) [draw,matrix of math nodes] at ($(mB.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \draw[dashed](mA.north east)--(mC.north east); \draw[dashed](mA.north west)--(mC.north west); \draw[dashed](mA.south east)--(mC.south east); \end{tikzpicture}   \end{document} Interpretation of ROC curve I have a ROC curve like the figure below:   What can be inferred given this kind of curve in general?  What are the differences of states 1, 2 and 3 in the figure?  What causes the diagram to jump straight from state 1 to state 3? Most of the ROC curves on the internet look like the figure below: Stack Overflow technology makes me write bad answers Today I had some time and felt like writing a relatively good answer on Stack Overflow; explaining both the underlying mechanisms and good practices, unlike usual declarative answers in a "do that" style.  It took me like an hour of time, $50 if I were free lance. It'll bring me nothing but 3-5 occasional views, and - if I get lucky - some ignorant comments.  You know, it's quite discouraging.  Instead of pride for the job that was done, I feel frustrated. I know, this answer may offer help to the OP, if he ever notices it (being already satisfied with other answers).  So the only fate the answer will meet is to sink forever, below thousands other questions, falling like snow.  So, being a programmer, I am thinking of efficiency. It's a waste of time to spend so much effort to answer one question when there is no way to [actually] reuse it. So, I wouldn't do it again but just turn back into short answers mode without explanations.       Seeing other answers, I don't think I am only one with such feeling.  The current policy is driving knowledgeable and educational users away, yet it attracts inexperienced ones in numbers. Ones who will tell you to use [some technology] which they never have laid their hands upon. Ones who write answers just because they have learned some "good practice" from other questions, and this "knowledge" becomes an endless source of rep points.  The idea of Stack Overflow is rotten. It was excellent when trees were high and traffic was low.  It turned bad now; it has become a honeypot for all the "enthusiast programmers" of the world, eager to share their 2 cents faster than you can say the word "close".  You're killing a great resource. Despite all the nice words and proper declarations, it discourages the reuse of the knowledge. And encourages fast on-site answers.  Of course, the rules will tell you contrary. It's all nice and proper (if you don't mind it's strict mutual exclusivity). But mechanically, technologically it works in the way I described.  I tried to write a tag wiki. You know, it was biggest waste of time in my life. As a matter of fact, it is not a wiki at all; you can't even link to a certain section. Not to mention that nobody ever has the idea to read tag wikis.  Reference questions? Don't make me laugh. There is no good mechanism to store and categorize links. Favorites, although intended for the different purpose, are a chunk of crap when you want to use it as a reference. So, you can't compete with 30-seconds answers: a question will get 5 answers before you can even find an appropriate link.  Okay, I can organize links myself but, there are some problems:   I am just a human. I cannot remember everything I am just a human and I cannot answer questions at rate of 1 qpm. One person cannot fight the Tide. It's helpless. When technology makes people writing fast [incomplete] answers -  no good will of proper declarations can do a thing.  Even if a question gets finally closed, it won't hurt anyone - the hydra grew two more already.   You'll call me a brute for insulting honest users and innocent volunteers, but that's fact: There are much more inexperienced users than experienced ones. And inexperienced users have much more spare time to hang over than experts who have their job to do. So, from that fact we can conclude that most of the answers on Stack Overflow have a low quality. Nobody cares. But I do.   There should be a technical way to limit the activity of not-so-knowledgeable-yet-eager-to-answer users. There should be a technical preference for the closing a question over answering it (have a silly answer for the silly question? welcome to comments). There should be a technical solution to accumulate and structure knowledge, a system that lets you choose a good answer instead of writing it by hand.   That's the only way. Otherwise Stack Overflow will remain an automated dispenser of bad practices.  Well, I just felt like letting it out again. Thank you for reading. Whenever we have a changing magnetic flux through a conductor an induced electric field is generated. It is often said that these induced electric field lines are circular in shape.But why is this so?Why are these field lines not elliptical for example? How can this shape be mathematically derived?  Also does this have anything to do with equipotential surfaces?Do such non-conservative electric fields even produce a potential field?If not why? What's the most efficient way to select the last n rows in a table without changing the table's structure? What's the most efficient way to select the last n number of rows in a table using mySQL? The table contains millions of rows, and at any given time I don't know how large the table is (it is constantly growing). The table does have a column that is automatically incremented and used as a unique identifier for each row. How to select multiple files with keyboard only in windows explorer Kate forces itself to be the default application for HTML I have a dataset with a large number of Yes/No responses. Can I use principal components (PCA) or any other data reduction analyses (such as factor analysis) for this type of data? Please advise how I go about doing this using SPSS. How to import a module given its name as string? I'm writing a Python application that takes as a command as an argument, for example:  $ python myapp.py command1   I want the application to be extensible, that is, to be able to add new modules that implement new commands without having to change the main application source. The tree looks something like:  myapp/     __init__.py     commands/         __init__.py         command1.py         command2.py     foo.py     bar.py   So I want the application to find the available command modules at runtime and execute the appropriate one.  Python defines an __import__ function, which takes a string for a module name:     __import__(name, globals=None, locals=None, fromlist=(), level=0)      The function imports the module name, potentially using the given globals and locals to determine how to interpret the name in a package context. The fromlist gives the names of objects or submodules that should be imported from the module given by name.       Source:    So currently I have something like:  command = sys.argv[1] try:     command_module = __import__("myapp.commands.%s" % command, fromlist=["myapp.commands"]) except ImportError:     # Display error message  command_module.run()   This works just fine, I'm just wondering if there is possibly a more idiomatic way to accomplish what we are doing with this code.  Note that I specifically don't want to get in to using eggs or extension points. This is not an open-source project and I don't expect there to be "plugins". The point is to simplify the main application code and remove the need to modify it each time a new command module is added. So, through typing several commands I've found that there's not only ls, but l and la too. There doesn't appear to be any man entries on Ubuntu 12.14. They all appear to do similar things with minor differences:  $ ls app        config     CONTRIBUTING.md  doc       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script  tmp     VERSION CHANGELOG  config.ru  db               features  Gemfile.lock  lib        log      PROCESS.md      public    README.md  spec    vendor $ la app      CHANGELOG  config.ru        db   features  Gemfile       .git        Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   .rspec  .secret     spec  .travis.yml  VERSION .bundle  config     CONTRIBUTING.md  doc  .foreman  Gemfile.lock  .gitignore  lib        log      PROCESS.md      public    README.md  script  .simplecov  tmp   vendor $ l app/       config/    CONTRIBUTING.md  doc/       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script/  tmp/     VERSION CHANGELOG  config.ru  db/              features/  Gemfile.lock  lib/       log/     PROCESS.md      public/   README.md  spec/    vendor/   Just as a bit of trivia, are there more of these and what do they do? Is here any place to find this out? Unfortunately, google searching these commands gets ignored because they're so short. I'm unable to access the following website from my Internet Explorer 8 under Windows XP with Internet Explorer unable to display web page error. By the other hand the FireFox browser at the same Windows XP can access this website.  How to solve the issue with IE8 ? How do CSS triangles work? Which annotation should I use: @IdClass or @EmbeddedId The JPA (Java Persistence API) specification has 2 different ways to specify entity composite keys: @IdClass and @EmbeddedId.   I'm using both annotations on my mapped entities, but it turns out to be a big mess to people who aren't very familiar with JPA.   I want to adopt only one way to specify composite keys. Which one is really the best? Why?