Suppose $(a_n),(b_n)$ are two convergent sequences such that there exists some $N\in\mathbb{N}$ such that $a_n\geq b_n$ for all $n\geq N$. Then $$\lim_{n \to \infty} a_n \geq \lim_{n \to \infty} b_n.$$   This is an obvious fact but I'm not sure how to formalize the proof.  Take the subsequences $a_{n_k}$ and $b_{n_k}$ which are defined to be the $N$th tail of $(a_n)$ and $(b_n)$ respectively. Since every subsequence of a convergent sequence converges to the same limit, we have $$\lim a_n=\lim a_{n_k}\geq \lim b_{n_k}=\lim b_n.$$  Is there a better proof than this?
Theorem  Let $\{a_n\}$ and $\{b_n\}$ be convergent real sequences. Assume that there exists a $N\in\mathbb{N}$ so  $a_n\le b_n$ (eq. 1)  for all $n\ge N$. Then  $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$.  My attempt at proving  Let $a=\lim_{n\to\infty}a_n$ and $b=\lim_{n\to\infty}b_n$. $a$ and $b$ are real numbers because the limit of a sequence of real numbers is a real number (which is a corollary in my textbook). For proof by contradiction, assume  $b&lt;a$.  Let $\epsilon=\frac{a-b}{2}$. Then we can find $N_a,N_b\in\mathbb{N}$ so  $|a-a_n|\le\frac{\epsilon}{2}$ when $n\ge N_a$, (eq. 2)  $|b-b_n|\le\frac{\epsilon}{2}$ when $n\ge N_b$. (eq. 3)  In a textbook I found that I should do the following:  Choose a $n$ so $n\ge\max(N,N_a,N_b$), then eqs. (1-3) are satisfied. Then  $a-\epsilon\le a_n+\frac{\epsilon}{2}-\epsilon$  How do I get that last thing? Is it necessary to do a proof by contradiction or can one do a direct proof?
Why does the product topology allow proper subsets for only finitely many elements? Why is the work done on a charge calculated from infinity? Why is the work done on a charge calculated from infinity to a point? Why not from one particular point to other? $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$   by directly substituting $x=0$ i got $\infty$ by using L-H's rule, i got $-1/8$   the given options are  $a)\frac{24e}{11}$    $b)\frac{11e}{24}$    $c)\frac{e}{11}$    $d)\frac{e}{24}$    may be the question would be  $$\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e+\frac{ex}{2}}{x^2}=\,?$$ Why is saltatory conduction in myelinated axons faster than continuous conduction in unmyelinated axons? Reflectivity of a glowing-hot metal surface When a polished piece of metal (or steel in particular) is heated to incandescence, how do its reflective properties change?  Given a mirror-like surface, would the object temporarily cease to act like a mirror at a certain temperature?  Would the specular reflection of light turn into diffuse reflection; in other words, would the surface temporarily become matte/dull?  I am not asking about permanent changes to the object such as melting, but rather can there be a temporary change in the reflectivity only while the the object is hot, that disappears when the object is cooled down?  Background of the question: I have a laser scanner that usually has trouble scanning shiny surfaces with high specular/ low diffuse reflectivity; and it seems to work a lot better for hot steel pieces than for cold ones. Glowing steel also looks less shiny to me than cold steel, but of course that is hard to tell with all the glowing Solve for $x$ in $6x^2-25x+12+\frac{25}{x}+\frac{6}{x^2}=0$ I simplified to get $6x^4-25x^3+12x^2+25x+6=0$  Or $6 (x^2+1)^2+25x(1-x^2)=0$  Then I substituted $z=x^2+1$, to get $6z^2+25\sqrt{(z-1)}(2-z)=0$  I can't find a way to proceed further. Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples? What's the "Bad magic number" ImportError in python, and how do I fix it?  The only thing I can find online suggests this is caused by compiling a .py -> .pyc file and then trying to use it with the wrong version of python. In my case, however, the file seems to import fine some times but not others, and I'm not sure why.  The information python's providing in the traceback isn't particularly helpful (which is why I was asking here...), but here it is in case it helps:  Traceback (most recent call last):   File "run.py", line 7, in &lt;module&gt;     from Normalization import Normalizer Question on groups of order $pq$ Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ fontawesome icons are getting too big using XeLaTeX The following example must be compiled with XeLaTeX to reproduce the issue.  % arara: xelatex \documentclass{article} \usepackage{fontawesome} \begin{document} Text Text \faMobilePhone\ Text Text \end{document}   Normally I am using TeXworks to compile and view the output. With the internal viewer I am getting the desired result. The following image demonstrates it:    However if I open the pdf with Preview (viewer of Mac) the scaling of the icon fails.     If I open the pdf file with Adobe Acrobat Reader the icon is still scaled correctly. If I print the page my printer is also not able to scale the icons correctly.   If I use LuaLaTeX everything seems to work correctly.   Can you reproduce the issue and do you know a solution? iOS 9.2 double touch fingerprint sensor I was messing around this morning and found that if I quickly double touch (NOT double-click) the fingerprint sensor on my iPhone 6S+ (iOS 9.2), the current app (or the home screen) slides down to the bottom half-screen, showing a blurred home screen background on the top half.  It doesn't matter if I use a finger that the sensor recognizes.  It goes back to normal after about 8 seconds or if you double-touch again, but I can't figure out what this mode is for??  Is this a bug or some hidden feature I haven't heard about yet? Why is $1 in a function not printing the script's first argument? 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My work computer is a 21" iMac Core 2 Duo and it drives me crazy because I'll quickly drag-resize a window larger by pulling down and to the right but instead of staying locked vertically above the dock it'll slide behind it which I don't like so then I've gotta jiggle the window around to get it all back into visible space... my home computer (6 core tower) doesn't have this problem even though I've, from what I can tell, set it up nearly identically.  Both machines running the latest version of snow leopard.  Here's a video captured from my machine at home showing the window resize behaviour I prefer. If it helps at all I can do another one from my work machine tomorrow showing the windows resizing behind the dock. How do I install PowerShell? Not long ago I read something on Microsoft's website about PowerShell being available for Ubuntu.   How would I go about the install of PowerShell?  I'd appreciate guidance for both Xenial &amp; Trusty.   Any installation method is fine, whether compiling source or relying on a package-manager.  Source-compiling methods should contain beginner-level detail. What defines a PC's consciousness of self, memories and knowledge? Show that the boundary of a set equals the boundary of its complement $\newcommand{\bdy}{\operatorname{bdy}}$  I'm trying to show that $\bdy(A) = \bdy(A^c)$.  I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know where to go from there.  Any help or hints would be very much appreciated. Parse a Json Array in to a class in c#