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How to force a table into page width? I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Does a non-abelian semigroup without identity exist? Difference between cp -r and cp -R (copy command) I'm a web developer, started to become a web designer. However, I think my understanding of colors is not practical yet. There are tools out there that can give you color harmonies, and more stuff, like , but they don't give you the start point. I mean, how should I know that for this web site, I'd better use green and for that website, choose red. How should I know that?  I think color selection is more like an art, instead of being a science, and I'm like "you should become an color artist, in order to become a web designer".  How can I improve my color selection skill? Definition of "pepper" in hash functions Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.  For example: $32 = 21 + 8+3 = F_8+F_6+F_4$  Research effort:  The base step it's simple:  Let $k=1$ it can be britten as $k=1=F_2$  For the inductive step I considered:  Let $k = k  F_2 =F_2+F_2+\cdots+F_2 = \sum_{i=2}^w a_iF_i, a_i =\{0,1\} $  Then if $k$ suffice I want to see if $k+1$ suffices too... But I'm not realy seeing how to use the inductive hypothesis, so I assume It's wrong.  Any thoughts on which can the inductive step be? App version: 1.0.35 Device: Nexus 7 Problem: Users with weird/unusual name won't display on the Android app, there will just be a blank space where the username should be. For example,  user is showing as:   On a question:   On their profile: Bootstrap 3 - Why is row class is wider than its container? What are the complexity guarantees of the standard containers? How do I use reflection to call a generic method? What is the best way to share/distribute Excel macro? Why did the Nazgûl not attack a second time at Weathertop? What is another word for 'all round' capabilities? I am looking for a positive, passionate word to describe the all-round capabilities of a person.   Are there any words better than 'all-rounder'? Can anyone recommend a simple API that will allow me to use read a CSV input file, do some simple transformations, and then write it.  A quick google has found  which looks promising.  I just wanted to check what others are using before I couple myself to this API. Which repository source do I need to add to my sources.list to access the package firmware-linux-nonfree?  I am using Deb 6 testing 32 bit. In QGIS, I have been learning a few expressions to style my trail maps but still know little and am stumped on what to use to filter my symbol data. I am showing Parking locations on my map. I use a marker symbol as a background element, so I can do callouts. I created a data field and added values to display beside this symbol a number that references a list of parking addresses in the book's text. All good except if the field has no number to display, it is NULL, it does not display my marker symbol. I need to filter this, so it does show even when there is no value in the field. It is generally taught that  function as adjectives modifying a noun (or pronoun).  In a sentence such as:     A major accident occurred on our bus's route to school, making us late for class.   or     The gas tank exploded, sending shrapnel through the air.   are the participle phrases modifying their preceding clauses? (If not, what are they modifying?) Usually I use beamer  with the Warsaw theme which puts theorems (and other things) in boxes. Now I switched my theme to Marburg but i still want to keep the boxes. How can one do that? For $0 < x < \pi/2 $, prove that $x + \frac{x^{3}}{3} <\tan x $ I proved as follows:  Let $f(x) = \tan x -  x - \frac{x^{3}}{3}.$ Then $f '(x) = \sec^{2}x - 1 - x^{2}$ and $f''(x) = 2\sec^{2}x\tan x-x &gt; \tan x - x,$ since $\\tan x &gt; x,$ $f''(x) &gt; 0$ so $f'(x)$ is increasing. Since $f '(0) = 0,$ $f '(x) &gt; 0$ for all $x \in (0, \frac{\pi}{2}).$ So $f(x)$ is increasing. since $f(0) = 0,$ $f(x) &gt; 0$ for all $x \in (0, \frac{\pi}{2}).$  But my proof seems complex. Is there any simpler way to prove this?