The Distributive Law is that $(a+b)c = ab + bc$ and that $a(b + c) = ab + bc $. I'm wondering whether there is a proof of why it is true? I know it's a logical statement and a law, but Bertrand Russell and Alfred North Whitehead did prove that $1 + 1 = 2 $.
I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it.  Perhaps we should take some time to prove these "obvious" things. I feel like if I cannot prove it, then I do not fully understand it. I would really like a proof of this most basic law, because I have been pondering over it for quite a while. Thanks!  Edit: This proof is for real numbers.
Why do we use 'did' with questions using the simple past tense? How do I connect the initial highway to my six-lane road? Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.   I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$  I am trying to go ahead of this step. How would one create an iterative function (or iterator object) in python? Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes. Use  strong mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.    I know the steps of weak mathematical induction... Basis step: $p(n)$ for $n=1$ or any arbitrary $n_0$ ... show that it is true Inductive hypothesis: $p(n)$ for $n=k$ ... Assume that it is true for $n=k$ Inductive step: $P(n)$ for $n=k+1$ ... Show that this is true for $n=k+1$ Drush from GitHub stopped working Secure an Ubuntu OpenSSH server from Brute force attacks but without a firewall or SSH key pair? I use Ubuntu 16.04 and seek to harden my SSH authentication in a special manner.  The current situation:  I have a machine with a minimal Ubuntu server I use mainly to transfer files to via its local OpenSSH server. Now, I don't have a firewall on that machine from a few reasons and I also avoid using a key pair hence I use only a password. One of the only ways I have left to defend from brute force attacks, and the one I most desire at the moment, is using a mechanism that blocks a user for X amount of hours, after Y amount of connection tries.  The desired situation:  I wish to have a standalone mechanism (that is, not as part of a firewall) that blocks a user for X amount of hours, after Y amount of connection tries as a way to defend from brute force attacks.  My Question:  Do you know a utility (and specific configuration) that will let me achieve the desired situation? I wanted to send a single email to different users. I save an email to drafts and send the email to a user and it goes to the sent box instead of staying in the drafts box. Is there a way I can send an email repeatedly from the drafts box? Novel about a man who has his car stolen and is cryogenically frozen with his cat? So this is a bit of a memory game of my behalf, but I'm trying to remember the name of a (presumably) Sci-Fi novel with only a recollection of the following:   A man owns a (yellow?) car parked outside his house along with a cat The car goes missing or is stolen The man decides to be cryogenically frozen for 75 (fairly sure, could be 50?) years along with his cat (I might be muddling up novels but I'm sure there was a cat...?) He takes a walk along the street X years in the future when he wakes up and the currency he has in his pocket is now worthless   I don't think I got that far into the book and all the above happened relatively early.  I read it about 10 years ago and at a guess, it's an 80's novel - all of the above details might be inaccurate, but I'm certain about the cryogenic freezing and the currency aspects.  I'd be surprised if this one gets anywhere! 14.04 LTS No Dash / Launcher after update through Software Updater How to get only time from date-time C# Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.  So far, I have not found any precise definition in literature. How can I wrap text around a bottom-right div? Every time I try to do something seemingly-simple in CSS, it doesn't work.  I have a content div that contains a 460x160 image. All I want to do is position the image at the bottom-right corner and wrap my text around it.  &lt;div id="contents"&gt;     &lt;img src="..." /&gt;     text text text text text text ... &lt;/div&gt;   So I want it to look like:  ------------------ | text text text | | text text text | | text text -----| | text text |    | ------------------   Doing it with a top-left or top-right image is cake:  #contents img { float:right; }  ------------------ | text text |    | | text text -----| | text text text | | text text text | ------------------   Now how do I push that to the bottom? The best I've come up with so far are:  #contents img { float:right; margin-top: 90%} // really needs to be 100%-160px  ------------------ | text text      | | text text      | | text text -----| | text text |    | ------------------   In this case the text does not print in the margin, so there is white space above the image.  #contents { position:relative; } #contents img { position:absolute; right:0; bottom:0; }  -or-  // move the img tag under the text in the html and: #contents img { float:right; margin-top:-160; }  ------------------ | text text text | | text text text | | text text -----| | text text | te | ------------------   In this case the text prints over or under the image.  So... how can I accomplish this? I'd like to typeset documents using the '' of page layout. Good examples are on pages 40-53 of . The  package is helpful for this, but two additional features would be useful:   Automatic flow of text across (arbitrarily placed) columns / textpos blocks. Vertical line spacing enforced to be an integer multiple of a base line spacing, even when interrupted (for example) by an equation of arbitrary height.   Any ideas as to how either or both of these could be accomplished? What happens to the second Marty at the end of Back to the Future? In  (the original movie), Marty comes back from 1955 10 mins early to prevent Doc from being shot (which he fails at). He then rolls up to the Lone Pine Mall and sees himself hit 88 mph and shoot back to 1955 (again). Marty then carries on with this life as usual. What happens to the Marty that goes back to 1955 at the end of the film?  This obviously is a recursive problem where it would create another timeline and Marty would keep coming back 10 mins early to view another Marty zip back to 1955? Is there a way to search the latest version of every file in TFS for a specific string or regex? This is probably the only thing I miss from Visual Source Safe...    Currently I perform a Get Latest on the entire codebase and use Windows Search, but this gets quite painful with over 1GB of code in 75,000 files.    EDIT: Tried the powertools mentioned, but the "Wildcard Search" option appears to only search filenames and not contents.  UPDATE: We have implemented a customised search option in an existing MOSS (Search Server) installation. Vertical spacing in floats I am using a float inside the text. As I found out, using \intextsep I could set space above and below the float and the texts. But when I am using \setlength{\intextsep}{5mm}, the space below the float is more than the one above it. Why they are not equal?  % Preview source code  %% LyX 2.0.0beta1 created this file.  For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{report} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \setcounter{secnumdepth}{3} \setcounter{tocdepth}{3} \usepackage{graphicx}  \makeatletter  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% A simple dot to overcome graphicx limitations \newcommand{\lyxdot}{.}   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{setspace} \setlength{\intextsep}{5mm} \setlength{\belowcaptionskip}{0mm}  \setstretch{2}  \makeatother  \usepackage{babel}  \begin{document} Text Text Text Text Text Text Text Text Text Text Text Text  \begin{figure}[h] \begin{centering} \includegraphics[scale=0.5]{\lyxdot \lyxdot /Pictures/fedora} \par\end{centering}  \caption{}  \end{figure} Text Text Text Text Text Text Text Text Text Text Text Text \end{document} Why does the following code sometimes causes an Exception with the contents "CLIPBRD_E_CANT_OPEN":  Clipboard.SetText(str);   This usually occurs the first time the Clipboard is used in the application and not after that. I want to switch to French locale. So I tried the below command:  myUbundu@myUbundu-desktop:~$ export LC_ALL=fr_FR   But I am getting the warning  -bash: warning: setlocale: LC_ALL: cannot change locale (fr_FR)   How to set the locale to French? Do I need to install additional packages? Does "non-" prefixed to a two word phrase permit another hyphen before the second word?  If I want to refer to an entity which is defined as the negation of another entity by attaching "non-" it seems strange to attach the "non-" only to the first word when the second one is really the word naming the entity. For example,     non-control freak   looks like a person obsessed with not being in control rather than one who is simply not obsessed with control.     non-control-freak   actually looks better because the "freak" is attached to the "non-" as much as it is to "control-", without the space implying the presence of a phrase break.    and its answers make it sound like totally detaching the "non" could be an option, but if I were following the convention of hyphenating "non-" could I add a hyphen after the next word to resolve this problem?