I know I am supposed to prove this using contradiction, but I am not sure how to proceed.  Here's what I have thus far:  Assume that there exists a smallest rational number $x$ such that $x &gt; 1$.  $X = \dfrac{a}{b}$ where $a$ and $b$ are integers.
How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
There is no such thing as magnetism? Here's an interesting "proof" that there is no such thing as magnetism. I know the answer but I love this so much I had to ask it here. It's a great way to confuse people!  As we all know, $$\nabla \cdot\vec{B} =0$$ Using the divergence theorem, we find $$ \iint_S \vec{B} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{B} \, dV = 0$$ Since $\vec{B}$ has zero divergence, there exist a vector function $\vec{A}$ such that $$\vec{B} = \nabla \times \vec{A}$$ Combining the last two equations, we get $$\iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Applying Stokes' theorem, we find $$\oint_C \vec{A} \cdot \hat{t} \, ds = \iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Therefore, $\vec{A}$ is path independent and we can write $\vec{A} = \nabla \psi$ for some scalar function $\psi$. Since the curl of the gradient of a function is zero, we arrive at: $$\vec{B} = \nabla \times \nabla \psi = 0,$$ which means that all magnetic fields are zero, but that can't be!  Can you see where we went wrong? Range of $(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$ is If $a_{1},a_{2},a_{3},a_{4}\in \mathbb{R}$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$ and $a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4}=1.$ Then Range of $$E =(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ is  Try: From $$(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ $$=2(a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4})+2(a_{1}a_{2}+a_{2}a_{3}+a_{3}a_{4}+a_{4}a_{1})$$ $$=2-2(a_{1}+a_{3})(a_{2}+a_{4})=2+2(a_{1}+a_{3})^2\geq 2$$ and equality hold when $\displaystyle a_{1}=-a_{3}$ and $a_{2}=-a_{4}$ Could some help me how to find its upper bound, Thanks I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$. Using Fermats prime numbers to prove that there is infinitely many prime numbers I installed Ubuntu alongside Windows but I did not disable secure boot as someone told me it could cause problems with boot loader. So when installation finished and I rebooted, the system went straight into Windows. I went to Ubuntu IRC Support and they told me to turn off the secure boot and reinstall Ubuntu. I did that but my system still boot into windows.  This is a picture of partitions on my system:    This is my first Ubuntu install on my new laptop. I followed everdaylinux guide to install it and I used something else option. I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can someone give me a hint? I really appreciate     Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$ How can I emulate 1.7.10's adventure mode in 1.8? "C" sizeof with a type or variable Probably a novice question, but how do I correctly diagnose from a photo for what reason that camera wasn't able to focus properly?  It can be any 1 of the following   Unsteady hand Lens issue Metering Re-Calibration required in camera or ...   Is there a set of unsaid protocols that needs to be followed by a photographer?  Goal is to get a sharp and crisp image prior to editing.  [PS: I own a Nikon D5200.] Why is "idea" sometimes pronounced as "idear"? I know that idea is pronounced as /aɪˈdiə/, but I've meet several people in real life who put an 'r' at the end of the word. How come? How to treat differentials and infinitesimals? In my Calculus class, my math teacher said that differentials such as $dx$ are not numbers, and should not be treated as such.  In my physics class, it seems like we treat differentials exactly like numbers, and my physics teacher even said that they are in essence very small numbers.   Can someone give me an explanation which satisfies both classes, or do I just have to accept that the differentials are treated differently in different courses?  P.S. I took Calculus 2 so please try to keep the answers around that level.  P.S.S. Feel free to edit the tags if you think it is appropriate. Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?  Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and constrained only by the Newtonian gravity of the $N$ particles with no external gravity.    My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical? Will my device work with Ubuntu? I have a device on which I would like to install Ubuntu. How do I find out if Ubuntu will work on it? Repost it with a minimal working example for the amsmath must be loaded earlier than mathspec problem Invisible space when trying to remove file (but exclude all files in a list)? I'm trying to accomplish a simple task using the excellent post : I would like to remove all files from a folder except those listed in a .txt file contained in the folder. The syntax works fine, but for some reason all files are removed - basically the script is saying that there are no correspondences between the filenames in the folder and those in the .txt file - but I can see that there are!   I thought maybe there might be some invisible white space after my file names that I cannot see (in the folder or the .txt file). But I don't think so...Is there another solution that I am not thinking of? Definition of $\mathbb{R}^\infty$ Question: Why is the topological space $\mathbb{R}^\infty$ defined to be the subset of $\prod_{i=1}^\infty \mathbb{R}_i$ consisting of sequences $(a_i)_{i=1} ^{\infty}$ such at most finitely many $a_i\neq 0$? Why does one insist on the condition that $a_i\neq0$ for at most finitely many $i$? How can I match a quote-delimited string with a regex? If I'm trying to match a quote-delimited string with a regex, which of the following is "better" (where "better" means both more efficient and less likely to do something unexpected):  /"[^"]+"/ # match quote, then everything that's not a quote, then a quote   or  /".+?"/   # match quote, then *anything* (non-greedy), then a quote   Assume for this question that empty strings (i.e. "") are not an issue. It seems to me (no regex newbie, but certainly no expert) that these will be equivalent.   Update: Upon reflection, I think changing the + characters to * will handle empty strings correctly anyway. Let's say that I have a folder, within my Documents folder, that has files I want no one to access without a password.  Is there a way to lock down that folder so that it's password protected / encrypted?  In fact, is it possible to password-protect any individual folder? "the better" vs "the best" of two players $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$