Cauchy's induction principle states that:     The set of  propositions  $p(1),...,p(n),...$ are all  valid if:           $p(2)$ is true.   $p(n)$ implies  $p(n-1)$ is true.   $p(n)$ implies $p(2n)$ is true.      How  to prove  Cauchy's induction principle? Can we use it to  prove what we  can prove with weak  and strong induction?  If yes how  to  prove using Cauchy's induction principle  $$ 1+2^1+2^2+...+2^n=2^{n+1}-1 $$
I was reading a blogpost here:  () One thing that threw me off was that after the first four large displayed equations, there is the statement &quot;it is sufficient to prove that if the theorem holds for $n=m+1$, then it holds for $n=m$.&quot; How is this type of induction valid? I browsed around for things like backward induction, reverse induction, and Cauchy induction, but couldn't find a justification for how this is valid. With the usual forward induction of verifying a base case and proving $P(n)\implies P(n+1)$, it's easy to intuitively understand how induction will show a property holds for all natural numbers (or at least starting at the base case). But with this reverse induction, it seems to me that if you prove $P(m+1)\implies P(m)$, then if you were able to verify a specific case like $P(15)$, then you would only know it's true for numbers up to $15$. How does it actually prove the property for all naturals?
Big-O proof showing that t(n) is O(1) Let t(n) = $\sqrt{n^2+100n} - n$  Show that $t(n)$ is $O(1).$  I tried solving this using the formal definition of O( ) but am not getting anywhere. Quite confused on how to tackle this question. I have a DisplayPort based monitor, and a laptop with a DP out.  I recently got a second laptop with HDMI out, and went looking for a HDMI -> DP converter.  I was very surprised to find that such a thing effectively doesn't seem to exist.   and the  seem to imply that it's just not possible to go HDMI -> DP, without a very expensive converter box (not just an adaptor).  However, I can't find anything that explains why this is the case.  Is DP designed to be backwards compatible but can't interpret HDMI?  Is it a case of HDMI-quality signal not being high enough quality to be convertible?  Some other reason?  Note that this question is not looking for an adaptor - I'm looking for the underlying cause of the lack of them. Is there offside when the ball deflects off an opponent? I was refereeing a 10 year old's junior game today and this situation happened. The home team's defender attempted to  pass from their own half to their forward who was in an offside position beyond the halfway line.  I was going to blow the whistle but the ball deflected off an away defender and then went to the originally offside attacker who put the ball wide.  I viewed this as onside at the time because the ball struck the away team player, however now I think it was offside because at the time of pass, the home player was offside.  Is this offside or not? How do I blend multiple photos to remove random people from the background? I'm very excited about the new  for Android/iPhone. Let's say you're taking a picture of your friend in front of a landmark with several people walking by. The app allows you to remove the people from the photo, leaving only the background and your friend. The app isn't just doing a clone of the surrounding image to paint over unwanted subjects. It's actually taking multiple images, detecting which objects are moving, and letting you select them for removal. This sounds awesome! But I don't want to be limited to taking pictures with a smart phone.  I would like to take multiple photos of the same scene. (Maybe with a tripod, but it would be better if I didn't have to use a tripod.) Then I would somehow blend them together to remove unwanted elements. How do I do that? I know there are tons of photoshop tutorials out there, but all the ones I find just tell you how to use the clone tool. What are reaching fouls in the NBA? When a player tries to steal the ball from a dribbling player, the referees sometimes award a "reaching foul". I just can't see any difference between a steal and a foul Process Unicode files with BOM correctly with POSIX tools Trying to use grep today, I ran into the  of the  in a Unicode file (UTF-8, in this case). Specifically, I was trying to find a file beginning with XYZ with the pattern grep '^XYZ', but of course grep treated the BOM as three separate characters and did not match the first line of the file if the first line started with XYZ. I even tried to update the regular expression to ignore spaces ('^[[:space:]]*XYZ'), but to no avail.  Other questions have dealt with  or , but I want to know if POSIX tools have a general option to handle Unicode files correctly. If grep handled the Unicode file correctly, it would consider the file contents to start after the BOM and match XYZ on the first line just like any other line. Who is the Maker? C-3PO frequently speaks of “the Maker.” Is this a true droid religion, or is this a religious reverence by droids for their specific creators (in this case, Anakin, even though C-3PO has forgotten who he is)? Is this even a droid thing or is it just C-3PO? What else do we know about this “Maker”-worship? I'm trying to do a volumetric light on Cycles, and I've followed tutorials, but nothing has worked yet. How can I make a proper volumetric light in Blender, like the following one?    But to leave the light in the middle, like, almost a star, is this possible? I'm at the point to use an image plane besides the light which would be better... I need to use it moving the camera.  I'm actually trying to do this because I want to simulate the brightness of the Sun.  I tried the effect, but it still gave me an error: Why can Arwen decide her mortality? Why shouldn't root be allowed to login via ssh? why should root access be disabled for ssh? I always hear that's for security reasons. But I don't get it. What's different to login as non-root and then sudo su -? What's the preferred way to handle tasks that need privileged permissions?  Thanks  Thomas Missing Schengen entrance stamp Last night I took a night train from Romania to Hungary. I shared a compartment with another American backpacker (and others.) The border crossing on the Romanian side was normal -- all of our passports were stamped and the border officer recorded our data in his notepad. On the Hungarian side, some sort of commotion broke out further down the train car while we were being processed. The officer who was scanning and stamping our passports left and never came back. As a result, the other American backpacker never had her passport scanned or stamped.  What sort of difficulties might she run into since her passport has not been recorded as entering the schengen zone, nor does she have an extra stamp? What might get options be now, from within Schengen, to get her entrance recorded?  (When I say "scanned", the officers had hand-held devices that read the passport biometrics and I assume, recorded some sort of data somewhere.) Do spice containers need to be hermetic? When do you use the "this" keyword? I was curious about how other people use the this keyword. I tend to use it in constructors, but I may also use it throughout the class in other methods. Some examples:  In a constructor:  public Light(Vector v) {     this.dir = new Vector(v); }   Elsewhere  public void SomeMethod() {     Vector vec = new Vector();     double d = (vec * vec) - (this.radius * this.radius); } How can I get help on terminal commands? I often see things like     read man xyz   For more information read xyz's man page.   see info xyz    so I wonder:   How can I get help on terminal commands generally? What exactly are these man and info pages?   How can I find and view them? Beamer to HTML using make4ht \documentclass{beamer} \usetheme{Madrid} \begin{document} \begin{frame} aaa \end{frame} \begin{frame} bbb \end{frame} \end{document}   The above tex file is to be converted into html. I am using the following command   make4ht filename.tex   However it throws error. The basic question is "Is it possible to convert beamer presentation to html using make4ht?" From what I understand, the  states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon.  However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics.  Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions?  What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying  because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation! Is it possible to have two Win 10 user accounts running simultaneously, with one used locally and one used via RDP? I need to allow someone to remotely connect to my windows 10 computer, and am wondering if this is possible:  I have two user accounts right now on my computer, one for me to work on and one for my friend remotely. I want to use my user account locally for myself, and I want to allow him to RDP and log into the other account when he needs to work without disturbing my computer experience.   In other words, I don't want to have to stop what I am doing and log off of my user account when he needs to access his account remotely. Can two user accounts run simultaneously in this regard, with one being connected to via RDP?  If so, can you please explain how to set it up properly?  Thanks for your help! How to check if a number is a power of 2 Today I needed a simple algorithm for checking if a number is a power of 2.  The algorithm needs to be:   Simple Correct for any ulong value.   I came up with this simple algorithm:  private bool IsPowerOfTwo(ulong number) {     if (number == 0)         return false;      for (ulong power = 1; power &gt; 0; power = power &lt;&lt; 1)     {         // This for loop used shifting for powers of 2, meaning         // that the value will become 0 after the last shift         // (from binary 1000...0000 to 0000...0000) then, the 'for'         // loop will break out.          if (power == number)             return true;         if (power &gt; number)             return false;     }     return false; }   But then I thought, how about checking if log2 x is an exactly round number? But when I checked for 2^63+1, Math.Log returned exactly 63 because of rounding. So I checked if 2 to the power 63 is equal to the original number - and it is, because the calculation is done in doubles and not in exact numbers:  private bool IsPowerOfTwo_2(ulong number) {     double log = Math.Log(number, 2);     double pow = Math.Pow(2, Math.Round(log));     return pow == number; }   This returned true for the given wrong value: 9223372036854775809.  Is there a better algorithm? Convergence of Ratio Test implies Convergence of the Root Test In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... Canonical URL for a home page and trailing slashes