Prove a given distance function is in fact a metric: $ d(f,g) = \sup\limits_{x\in X}|f(x)-g(x)| $
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
Why does the jquery change event not trigger when I set the value of a select using val()? I can't get into my field research prize Prove that the subsequence of a convergent series with nonnegative terms converges Turn off automatic preview in apple mail I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons.  So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again!   But if it’s all virtual photons, how do we get the difference between a magnetic and electric field?  [This question needs the tag "magnetostatics" as well]. Possible Duplicate:          If I am using the word model in the context of financial models and the UK, then which of these words should I use? Also, are there any key difference in the meanings of the two? Would the One Ring have stayed hidden if Bilbo hadn't turned up? Question list on profile page not vertically aligned There's something wrong with the vertical alignment on the :   (click to enlarge)  On the , which uses a similar list but without the Favorite column, the vertical alignment is correct.  (Tested in the latest Firefox, Safari and Chrome on macOS Catalina.) javax.faces.application.ViewExpiredException: View could not be restored Install macOS Sierra on VirtualBox? I’m trying to install macOS Sierra on VirtualBox, but can’t seem to make it work. For reference, here are the steps I took.   Downloaded the Install macOS Sierra app from the App Store. Used the inbuilt media creation executable, CreateInstallMedia on a disk image. Created a new VM with the default settings for an OS X El Capitan, plus 128 VRAM. Since VirtualBox refused to accept the DMG, I tried splicing out the InstallESD. VirtualBox now accepted it, but wouldn’t boot. I tried to boot through rEFInd, but still no luck. I tried to patch it both ways (base system and ESD) with iesd. No luck. When booting to GParted LiveCD, it flat-out refused to discover any of the installer CDs.   How can I patch the installer to work with VirtualBox? I don’t want to use pre-patched versions of either the installer or an existing installation, as though to avoid piracy. I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$? What is the name of this connector from a fan? I am EE hobbyist and I often used recycled parts in my projects. I came across this connector to a Heat Sink Fan and would like to know the name of it.  Thanks for the help. Alternative set theories This is a (soft!) question for students of set theory and their teachers.   OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student encounter (at least to the extent of knowing that the theory exists, and about why it is thought interesting)?  Suggestions might be SP (Scott-Potter), NBG, NF, ZFA, IST, ETCS (for a few sentences of explanation, see   But what would be on your list of alternative set theories would be it good for advanced students to hear about, if only briefly? Why is the notation $\frac{\partial f}{\partial x}$ used? \overline outside of math mode If I want to draw an overline over some text, not an equation, how can I do it? \overline seems to only work in math mode, and when used outside of math mode, it automatically says "Missing $ inserted". How can I clone an ArrayList and also clone its items in Java?  For example I have:  ArrayList&lt;Dog&gt; dogs = getDogs(); ArrayList&lt;Dog&gt; clonedList = ....something to do with dogs....   And I would expect that objects in clonedList are not the same as in dogs list. Problem starting Gnome terminal This is the first part of the first half of the a proof in the Axiom of Choice. The proof that the Axiom of Choice is equivalent to the choice function. REsources: Keith Devlin's Joy of Sets: pg. 58. F.R. Drake and Singh Intermediate Set Theory: pg. 110).   I am confused on proving disjointness and will ask further questions of proof once I get this first part v. clear for myself.   AC1: Let $\mathcal F$ be a family of pairwise disjoint, non-empty sets. Then there is a set $\mathcal M$ that consists of precised one element from each member of $\mathcal F$  AC2: For every family $\mathcal F$ of non-empty sets, there is a choice function $\mathcal f$. (I.e. $\mathcal f(X)$ is a member of X, for each X in $\mathcal F$.   Disjointness: Two sets are disjoin if Q $\cap$ R = $\emptyset$. Meaning that they have zero elements in common. These sets do not overlap.   Show: AC1 $\leftrightarrow$ AC2   (1) ($\rightarrow$) Let $\mathcal F$ be a set of nonempty sets.   (2)For each X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X}) = {(x, X) | x $\in$ X})  (3) Show Disjointness.  An outline of how book proves  disjointness:  (a) if u = {(X $\times$ {X}) | x $\in$ X} then u is of the form $\lt  z, t \gt$ where z = X and t=x, i.e. x $\in$ X.   (b) then if u $\in$ ({X} $\times$ x) $\cap$ ({Y} $\times$ Y) $\rightarrow$ u = $\lt z, t \gt$ with z = X and z=Y, i.e. X=Y. Which means ({X} $\times$ x) = ({Y} $\times$ y).  Questions: (1) Apparently we cannot prove disjointness in $\mathcal F$ but must prove in a different set. If so, why is that?  (2) why is the case that if u $\in$ ({X} $\times$ X) $\cap$ ({Y} $\times$ Y) then z =X and z=Y? I am not following that step at all.   (3) why does the axiom of choice have to do with ordered pairs? Why are we setting X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X})?    Thank you! Is there a word or phrase for something that one might wish exists, but most certainly doesn't? Creating surface from points in ArcGIS for Desktop? how to create surface from points in ArcGis? I have a points (centroids of the neighbourhood and number of people living in every neighbourhood) and i would like to create surface that would present density of population.