I am about teaching a course on inequalities to secondary school children and I find we'll have to use at some point that if $$0&lt;a&lt;b,$$ then $$\sqrt a&lt;\sqrt b.$$  Therefore, I've been thinking of how one would go about explaining why this is true to these children, and the only way I know uses real analysis -- and they've definitely taken no calculus yet.  Hence, do you know of a way to prove this result using elementary means?  Thank you.
I've shown the existence of unique square roots of all positive rational numbers, so now I want to prove that the square root is monotone:   $0&lt;a&lt;b$ if and only if $\sqrt{a} &lt; \sqrt{b}$
Problem  Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. Show that $H$ is a proper normal subgroup of $G$.  So basically the result of this problem is that every group of order $2n$, with $n$ odd, is not simple.   The morphism $\phi$ can be seen as the action of $G$ on itself by left multiplication. Since $A_{2n} \lhd S_{2n}$, then $H=\phi^{-1}(A_{2n}) \lhd G$. I am stuck at showing that $H \neq \{1\},G$. All I could think of is that $G$ has an element $g$ of order $2$, then $\phi(g)$ is of order $1$ or $2$, but it is easy to see that $\phi(g)=1$ if and only if $g=1$, so $ord(\phi(g))=2$. I don't know which are the elements of order two in $S_{2n}$. I would appreciate some help to solve the problem, thanks in advance. What is the recommended wording for a generic error message Quantum levitation, flux pinning -- basically, when a superconductor floats above a permanent magnet, is really fascinating. But does the strength of the magnetic field influence the superconductor's height? And what relationship/formula suggests this?  For example, if I had a coil wrapped around a metal object and then ran a small current through it, a small magnetic field would be generated and a superconductor exposed to it would float about it at some height that I put it on.  But what if I increased the current, so that the magnetic field also increased? What happens to the superconductor? Does its height change? Or does it just get harder to "pull out" of its levitating state? Determining relationship (e.g. number of removes) with cousin? How do I figure out if someone is a first cousin, second cousin, cousin once-removed, etc? Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! Define ≡ in this situation? Is there a easy way to test ubuntu on new notebooks before buying it? What does negative R-squared mean? Let's say I have some data, and then I fit the data with a model (a non-linear regression). Then I calculate the R-squared ($R^2$).  When R-squared is negative, what does that mean? Does that mean my model is bad? I know the range of $R^2$ can be [-1,1]. When $R^2$ is 0, what does that mean as well? Is there any way to distribute spacing equally in every object?    Is there any option to do? I want 2MM spacing in every geometrical shape.  Thanks in advance. Can US stocks list on one stock exchange but trade on other US stock exchanges? I am trying to understand how the US stock market works. From my understanding, there is one stock exchange where a company is &quot;listed&quot;. This stock exchange listing requires the company to adhere to the regulations of the stock exchange where it is listed. Once a company is &quot;listed&quot;, its shares can start to trade on any stock exchange. For example, once a company lists on NASDAQ, its shares can be traded on NASDAQ, NYSE, Boston Stock Exchange, Philadelphia Stock Exchange, National Stock Exchange, EDGX Exchange, EDGA Exchange, NYSE Arca, etc. Is my understanding correct? Is it true that once a stock is listed on a US stock exchange, the stock can then trade on any US stock exchange? Consider the Level 2 quotes for Microsoft (NASDAQ: MSFT). Microsoft is only listed on NASDAQ, but it also trades on the NYSE and other stock exchanges:  My questions are:  Does this mean that companies only need to be listed on one stock exchange in order for their stock to trade on other stock exchanges?  If so, do companies need to adhere to the &quot;listing requirements&quot; of all the stock exchanges where their stock is traded, even though they are not &quot;listed&quot; on those stock exchanges? Does Microsoft have to follow the listing requirements of the NYSE even though it is not listed there?  If stocks can trade on any US stock exchange regardless of which US stock exchange the stock is listed on, why don't NYSE-listed companies reduce their yearly listing fees by listing on NASDAQ instead? At some point I accidentally ran off a cliff with my War Horse which resulted in a death. After respawning, I noticed my War Horse was gone and the default Tennessee Walker was my main horse.   How did the War Horse disappear and is it possible to retrieve it? I understand that multivariate regression models contain more than 1 dependent variable, but what is the difference between running a multivariate regression with dependent a and b to a set of independent variables, rather than running two regression models, one for a and one for b against the same set of independent variables? Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not. Say that a topological space is CH if it is both compact and Hausdorff.   Let $T$ and $T'$ be two topologies on the same set X that are comparable   but different, i.e., $T$ is either strictly smaller or strictly larger than $T'$.   Assume that $(\text{X}; T )$ is CH. Prove that $(\text{X}; T')$ is not CH.   I just can seem to get started, so I was wondering if anybody could give me a hint...  If $T' \subsetneqq T$, then the identity $id: (\text{X},T) \rightarrow (\text{X}, T')$ is continuous and surjective. This implies that $id(\text{X},T) = (\text{X},T')$ is also compact. So in order to show that $(\text{X},T')$ is not CH, I need to show that it is not Hausdorff, right? But I'm stuck... Encode/Decode URLs in C++ Does anyone know of any good C++ code that does this? Why do I need to type `./` before executing a program in the current directory? While executing a C program, a.out, using the Ubuntu terminal, why do I always need to type ./ before a.out, instead of just writing a.out? Is there solution for this? How can one cite theorems of references? Where can I find information about upcoming special issues in a field of study? How to implement Enums in Ruby? What's the best way to implement the enum idiom in Ruby? I'm looking for something which I can use (almost) like the Java/C# enums. Sorry for the bad English. I'm trying to use the tikz-cd package to make commutative diagrams. My code  \documentclass{article} \usepackage[brazil]{babel} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} A \ar[r, "x"] &amp; B \end{tikzcd} \end{document}    The problem occurs when putting a name under the arrow, the syntax "x" is incompatible with the brazil option of the babel package. When exchanging brazil for english it works normally. I already tried using  \usepackage[main=brazil, english]{babel}, \usepackage[brazil, english]{babel} e  \usepackage[brazil, main=english]{babel}   But it did not work. I also tried to use  \usepackage[english, brazil]{babel} \usepackage{polyglossia} \setmainlanguage{brazil} \setotherlanguages{english}   And finally  \selectlanguage{english}   However, it only works if I remove the option brazil titlesec: loss of section numbering with the new update (2016/03/15) After reinstalling TeXLive distribution (install-tl-20160318, Ubuntu 14.04), a issue occurs with the updated version of the titlesec package: the section numbering is lost.  What could it be missing? Is there a modification I have not seen?  Below is a MWE comparison of the two versions. The prevtitlesec package is a reinstall of the previous version (2011/12/15).  \documentclass{book} \usepackage[margin=2cm,a4paper]{geometry} %\usepackage{prevtitlesec}% 2011/12/15 version \usepackage{titlesec}% 2016/03/15 version \usepackage{lipsum}  \begin{document}  \chapter{Title of the first chapter} \lipsum[1] \section{A test section} \lipsum[4] \subsection{A test subsection} \lipsum[4] \subsection{A test subsection} \lipsum[4]  \end{document}   titlesec package 2011/12/15:    titlesec package 2016/03/15: