I am trying to prove this and have looked at similar questions to gauge how to approach this. I have: Suppose that there exists a smallest rational number greater than $\sqrt{3}$. We shall call that number $n$, which, as it is rational, can be expressed as $\frac{p}{q}$ $\frac{\sqrt{3}+n}{\sqrt{3}}$ is a number greater than $\sqrt{3}$ but less than $n$, but this number would no longer be rational now, would it?
I am working on a proof about Dedekind cuts on the positive rational numbers. I have been stuck for a while on the following point and would appreciate any help.  Given $x\in \mathbb{Q}^+$ such that $x^2&lt;2$, how to find $y\in \mathbb{Q}^+$ such $y&gt;x$ and $y^2&lt;2$?
Different mechanics of \hyperlink vs. \hyperref I am fiddling around with the hyperref package to build a hypertext structure into my pdf. At first glance, hyperref provides two similar macros to put links between parts of my document: \hyperlink{target}{text} and \hyperref[target]{text}. You might notice their slightly different syntax. What really interests me is their different working mechanics. I will illustrate this by the following example: \documentclass{article}  \usepackage{pdfpages} \usepackage[]{hyperref} \usepackage{mwe}  \begin{document}  \section{Section Heading} \label{sectionheading}  \includepdf[link=true, linkname=pdfpage]{example-image}   Link to section works with \hyperref[sectionheading]{hyperref} and with \hyperlink{sectionheading}{hyperlink} (but with slightly different targets on the page).  Link to pdf works with \hyperlink{pdfpage.1}{hyperlink} but not with \hyperref[pdfpage.1]{hyperref} (and not this way, either: \hyperref[pdfpage]{hyperref}).  \end{document}  To link to the labeled section heading, I may use both commands. While \hyperref directs my pdf viewer directly to the linked section, \hyperlink directs me somewhere slightly different, so I can see more context. To link to the \includepdf of pdfpages however, I may only use \hyperlink. \hyperref does not work at all. I suspect this is because of a subtle difference in their working mechanisms. While checking the aux file, I found a line \newlabel{sectionheading}{{1}{1}{Section Heading}{section.1}{}} which \hyperref seems to use as a target. However, I did not find any similar target for the included pdf. My question: I suspect, I understand the workings of \label and \ref very well. \hyperref seems to work quite similarly with the added clickability. Could you please explain to me how \hyperlink does work differently and how it is implemented? E.g. what mechanics explain the difference when clicked on the link and where and how are the link targets stored in the first place? Why does change in speed of a wave make it refract? When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract/bend?  I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this? Ways of getting a number with $n$ dice, each with $k$ sides Assume the dice are numbered from $1$ to $k$. My hunch is that this will form a normal distribution with a median at $n\cdot\frac{k}{2}$. However, I have no idea as to    turn this fact into an answer (I have a minimal knowledge of stats, but I know that I am missing the standard distribution) and this is probably the wrong approach   How can I approach and solve this problem? (Aside, this is not for a class, stats or other, so any and all approaches welcome).  *Edit: * I want to find the number of ways that the sum of the numbers that are rolled has a particular value, if $n$ dice are rolled, and each has $k$ sides, numbered $1$ to $k$. What words are commonly mispronounced by literate people who read them before they heard them? Number of reflexive relations defined on a set A with n elements When is a CDATA section necessary within a script tag? Are CDATA tags ever necessary in script tags and if so when?  In other words, when and where is this:  &lt;script type="text/javascript"&gt; //&lt;![CDATA[ ...code... //]]&gt; &lt;/script&gt;   preferable to this:  &lt;script type="text/javascript"&gt; ...code... &lt;/script&gt; Number of solutions to the congruence $x^q \equiv 1 \mod p$. Let $p, q$ be distinct odd primes, I would like to compute the number of solution $\mod p$ to the congruence $x^q \equiv 1 \mod p$. (Show that it's $gcd(q, p - 1))$  If $x^q \equiv 1 \mod p$, since $q$ is prime, this must mean that $x$ is either $1$ or has order $q$.   Hence, the problem reduces to finding the number of elements of $U_p$ which is either $1$ or has order $q$.  If $ q \nmid  p - 1$, then there is only one element, namely $1$.  If $q \mid p - 1$, then the number of elements of order $q$ is $\phi(q) = q - 1$, and so the number of solutions to the congruence $\mod p$ is $q - 1 + 1 = q$, and in either case the number of solutions is $\gcd(q, p - 1)$.  Does this look fine? How do I make a WinForms app go Full Screen I have a WinForms app that I am trying to make full screen (somewhat like what VS does in full screen mode).  Currently I am setting FormBorderStyle to None and WindowState to Maximized which gives me a little more space, but it doesn't cover over the taskbar if it is visible.  What do I need to do to use that space as well?  For bonus points, is there something I can do to make my MenuStrip autohide to give up that space as well? What would be the Slater's determinant representation for an excited state? Setup  Introducing this spinorbital notation:  \begin{align} \Psi_1=\chi_{(r1)}\alpha_{(\omega1)}  = 1 \\ \Psi_1=\chi_{(r1)}\beta_{(\omega1)}  = \bar{1} \end{align}  and the Slater's determinant, for a system of two electrons, is:  \begin{align} |\Psi\rangle = \frac{1}{\sqrt{2}} \Big|  \begin{array}{cc} 1 &amp; \bar{1}  \\ 2 &amp; \bar{2}  \\ \end{array}  \Big| = \frac{|1\bar{2}\rangle-|\bar{1}2\rangle}{\sqrt2} \end{align}  Question  Based on the example, what would be the Slater's determinant representation for an excited state like $\Psi_a^r$ (i.e. with parallel spins like: $|\bar{1}\bar{2}\rangle$) and $\Psi_{ab}^{rs}$ (feel free to use any number) ? I always thought with "any" I should use the plural, but on the internet I can find both:   It can be found in any book. It can be found in any books Do you have any books? It can be said in any language.  This can be understood by anyone.  It has been used in any form.   So, what's correct?  Is there any rule? Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. How do I share a folder on SAMBA? Why do people sometimes use the past perfect form of a verb when the past would suffice? (e.g. "you had mentioned") I noticed that my American friends tend to use the past perfect form when they use the verb mention in its past form, almost despite of the context of the sentence. I hear them say  "you had mentioned" a lot more often even in situations where the use of simple past would have been perfectly fine. Have you noticed it too and is it one of those unwritten rules, or is there a more formal explanation to it?  It seems like there was more verbs like that, but I am having a hard time remembering it now. If I do, I will edit the post and likely change the title to accommodate other verbs. How to cite a source that has vanished due to link rot? My colleague is writing a manuscript about a subject that is rarely reported in the mainstream literature (predation events). A few years ago on a fishing forum she found a photograph that demonstrated this phenomenon pretty well and saved it to her computer, but unfortunately between then and now the original source has vanished: Google Reverse Image Search can't locate it, the website is no longer running and is also unavailable in the Internet Archive. The original URL to the photo was also not retained.  What's the proper way to cite this source? It's not critical to include but it would definitely improve the final manuscript. Separator between command list and } Proof a set cannot contain its own powerset - Russell's Paradox How to show $f=g$ almost everywhere  Let two real-valued measurable functions $f$ and $g$ be such that for any measurable set $E$ the integrals $\int_E f\,d\mu$  and  $\int_E g\,d\mu$ coincide.  Could you please help.  It seems obvious. we could define the integrals as charges. can we do sth from here. but I do not know how to approach. Smooth transition between two materials How can I delete a (null) app in usage? In my usage list(settings>general>storage &amp; iCloud usage>manage storage), there is an app that registers as (null) with no icon, a version number that doesn't match anything I've previously downloaded, and it's taking up storage space. I first noticed it about a week or two ago. And no, I can't download recent apps to find the one that it is, because I downloaded multiple apps recently, and didn't notice the (null) app until my storage was critically low and I was looking to delete something. I am running iOS 9.2.1 on an iPhone 5C. How do I boot into a root shell? When booting to the graphical desktop fails we sometimes are recommended to boot into a "root shell" or to boot in "recovery mode".  How do I do that?