Recurrence relation: $x_{n+1} = \frac 12 x_n + \frac 1 {x_n},$
Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$
I'm not sure how to prove these basic theorems in propositional calculus. Instead of using the standard axioms, we're supposed to use:   Deduction Theorem (if $\Phi, \alpha \vdash \beta$ then $\Phi \vdash \alpha \to \beta$),  Reductio (if $\Phi, \alpha \vdash \, $, then $\Phi \vdash \lnot \alpha$), Cut Rule (if $\Phi \vdash \alpha$ and $\Psi, \alpha \vdash \beta$ then $\Phi \cup \Psi \vdash \beta$),  Inconsistency Effect (if $\Phi \vdash \, $, then $\Phi \vdash \beta$ for every formula $\beta$), and  the Principle of Indirect Proof (if $\Phi, \lnot \alpha \vdash \, $, then $\Phi \vdash \alpha$),    as all the axioms can be deduced using these theorems.  I don't really know how to start the proofs without using the axioms:  i) prove that $\lnot(\alpha \to \beta) ⊢ \alpha$  ii) prove that $\lnot\alpha \vdash \alpha \to \beta$  Any suggestions on how to start these proofs or any insight at all would be greatly appreciated!  Thanks! Can I pay two different costs with the same payment? Say I have a Zombie token, , and  on the battlefield. Can I sacrifice the Zombie token to activate both the Altar's and Seer's abilities?  Similarly, if I have one card in hand, and  and  on the battlefield can I discard the one card to both of their abilities?  If I have a  with 3 charge counters can I activate both of its abilities by tapping it to add a counter, and tapping it to remove 3 counters and destroy a permanent without untapping it in between? How to add shortcut keys for java code in eclipse Display date/time in user's locale format and time offset I want the server to always serve dates in UTC in the HTML, and have JavaScript on the client site convert it to the user's local timezone.  Bonus if I can output in the user's locale date format. I was wondering if $\mathbf{A}\times\mathbf{C}=\mathbf{B}$ and $\mathbf{A}\cdot{\mathbf{C}}= p$, is it possible to express C in terms of A, B, and p?  Note: A, B, C are vectors and p is a scalar.  Sorry if this question has been asked before. If so, can anyone give me a link to it? css background color with floating elements Deleted users who created a synonym are listed as "anon" in the tag synonyms list While browsing the  I came across the synonym [vote-to-close] -&gt; [closing] in which the creator is listed as &quot;anon&quot; — the username is not hyperlinked, just like how the user IDs of deleted users are listed in other places like in the list of close voters of a question, etc.  If this really were a deleted user, then I would have expected their user ID to be shown, and not &quot;anon&quot;. So, I checked using  for data on the tag synonym. And sure enough, it shows that the deleted user with the ID 147263 created the synonym. To double-check, I looked up the  and came across a bunch of synonyms created by &quot;anon&quot;. Checking with , these were all created by the now deleted user with the ID 80. Similarly, in the  there are several created by &quot;anon&quot;, but  says that these were created by the now deleted user with the ID 54262.  Feature request: Show the user ID of the tag creator under the &quot;Creator&quot; column in the list of tag synonyms if the user is deleted, to be consistent with how posts and actions by deleted users are attributed everywhere else. Why does C/C++ differentiates in case of array index out of bound #include &lt;stdio.h&gt; int main() {     int a[10];     a[3]=4;     a[11]=3;//does not give segmentation fault     a[25]=4;//does not give segmentation fault     a[20000]=3; //gives segmentation fault     return 0; }  I understand that it's trying to access memory allocated to process or thread in case of a[11] or a[25] and it's going out of stack bounds in case of a[20000]. Why doesn't compiler or linker give an error, aren't they aware of the array size? If not then how does sizeof(a) work correctly? A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse, two curves intersect in $4$ points. Converting MultiLinestring to Linestring with PostGIS I have a multiline string and want to convert it to a linestring in order to use some functions that only work with linestrings. I used ST_Dump() and got a collection of my geometries. If I merge them back together, I still get the multilinestring. Should I get the Npoints and combine them together or something, in order to end up with a simple linestring? ST_LineMerge() also does not work on my multilinestring, it returns the same. my example,  MULTILINESTRING((-3.16420835153456 55.9269166007097,-3.164222 55.926918),(-3.1642070167833 55.9269296196706,-3.16421351659546 55.9268662214904),(-3.16421351659546 55.9268662214904,-3.16421636372824 55.9268384509897),(-3.16421636372824 55.9268384509897,-3.16422182573761 55.9267851753802),(-3.16422182573761 55.9267851753802,-3.16422870102352 55.926718114886),(-3.16422870102352 55.926718114886,-3.16423309121073 55.926675293667),(-3.16423309121073 55.926675293667,-3.16423565148822 55.9266503211093),(-3.16423565148822 55.9266503211093,-3.16424103159897 55.9265978443265),(-3.16424103159897 55.9265978443265,-3.16424680776317 55.9265415044985),(-3.16424680776317 55.9265415044985,-3.16425267254583 55.9264843002995),(-3.16425267254583 55.9264843002995,-3.16425541048045 55.9264575949012),(-3.16425541048045 55.9264575949012,-3.16426111146586 55.9264019883556),(-3.16426111146586 55.9264019883556,-3.1642667032531 55.9263474469124),(-3.1642667032531 55.9263474469124,-3.16426957768543 55.9263194101362),(-3.16426957768543 55.9263194101362,-3.16427488261739 55.9262676666359),(-3.16427488261739 55.9262676666359,-3.16428009893088 55.9262167875066),(-3.16428009893088 55.9262167875066,-3.164282741107 55.9261910161221),(-3.1642875546472 55.9261440655823,-3.164282741107 55.9261910161221),(-3.1642875546472 55.9261440655823,-3.16429466890915 55.9260746741522),(-3.16429466890915 55.9260746741522,-3.16430092974527 55.9260136069079),(-3.16430092974527 55.9260136069079,-3.16430822838418 55.9259424170929),(-3.16430822838418 55.9259424170929,-3.16431547242401 55.925871759829),(-3.16431547242401 55.925871759829,-3.16431448732505 55.9258328901507),(-3.16431770120536 55.9257327846001,-3.16431547242401 55.925871759829),(-3.164339 55.925777,-3.16431770120536 55.9257327846001)) How to protect website from bulk scraping /downloading? Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$ I would change the font of section and subsection of my document. I have created it using book class. I would use this font:    According to Adobe pdf propriety, this font is AvantGarde-Demi. How can I use it in Latex. what command I should use to recreate this in latex. I think I should use titlesec package right? Help! My party isn't challenged? While I've been running my campaign, I'm starting to get stumped. I am playing with a party of four, and we recently switched from 3.5 to Pathfinder, but the problem was the same in both situations.  No matter what I throw at the party, they seem to have no problems. Even if I put them up against something that is supposed to be 'Epic' or 'overwhelming' in terms of CR, they make out just fine. The one exception to this is when I had several encounters in one day that taxed them to near death. Usually I have about two encounters a day, but that day I had four encounters, with two of them being APL+1 and the other two being APL+2, and that was challenging. But other than that nothing seems to faze them!  How can I make the game more challenging without making it harder?  Just for reference, the party is fairly standard, and low level (level 4):   Wizard (tends to resort to mind-affecting spells, or grease if the creature is immune) Cleric (heals a lot, but always has spells / healing left over) Rogue (sneak attack, of course, which is more potent in Pathfinder) Fighter (gets enlarged and in-charge'd) How to change Ubuntu 18.04 system fonts? I installed Ubuntu 18.04, and I want to customize the fonts. I have already change all the setting by tweak tool. But the system default font not be changed, is anything I need to do? How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? I have not much experience with ndiswrapper but some vague memory of using it with some broadcom module long back. Now, here is the problem. Realtek RTL8723DE wireless module is used in my new HP 15-BS576TX laptop. I haven't installed Linux yet. But, would like to check if Ndiswrapper can help use the Windows driver to work in Linux. Here is the Windows 10/8/7 driver for this RTL8723DE wlan module:   I extracted the exe and can see a folder /RTWLANE_Driver/Wind7X86/ which has rtwlane.sys and few other files which I presume is related to the wireless driver. Is there any chances of luck with ndiswrapper considering the driver executable actually contains win8/win7 drivers.  Thank you. Sometimes chicken breast can be thick and I never know how to tell when it's cooked! How can I be certain it is cooked all the way through? Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$. Prove if nonempty $K \subset \mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.   I manage to prove the existence of sup $K$ by using the Axiom of Completeness. But I failed to prove that $\sup K \in K$. Can anyone guide me ? Given $f_{n} : [a,b]\rightarrow \mathbb R$ is uniformly Lipschitz and $f_{n}\rightarrow f$, how can I show that $f$ is Lipschitz?  $f_{n}$ is uniformly Lipschitz $\leftrightarrow \exists K &gt; 0$ such that for all $n$ and for every $x,y\in [a,b]$ we have $|f_{n}(x)-f_{n}(y)| &lt; K|x-y|$.  $f_{n}\rightarrow f \leftrightarrow \forall~\epsilon&gt;0~~\exists~N$ such that $\forall~n&gt;N$ we have $\mid f_{n}(x) - f(x)\mid&lt;\epsilon~~~(=\mid x-y\mid?)$  Then we have  \begin{align} |f(x)-f(y)| &amp;\leq |f(x)- f_{n}(x)| + |f_{n}(y)-f(y)| + |f_{n}(x)-f_{n}(y)| \\ &amp;\leq |x-y| + |x-y| + K|x-y| \\ &amp;\leq (K + 2) |x - y| \end{align}  which shows that $f$ is Lipschitz. Can anybody tell me if this makes sense? Sorry if layout is poor. Displaying layout marks in a document set with different layouts Is there any package that allows for the display of layout marks within a document where the layout changes several times throughout? I'm using the geometry facilities to issuing \newgeometry commands throughout my document, but it doesn't update the displayed layout and neither does the showframe package or the facilities of the memoir class.  EDIT: As Werner has pointed out, the showframe option of the geometry package displays correctly any updated page layout set by issuing \newgeometry commands.   My trouble came out of the fact that I'm issuing also \pagecolor commands and the page color seems to be hiding the layout lines typeset according to the geometry package. My confusion only worsened because the showframe package layout lines were not hidden by the page color (the package got the layout wrong on some pages, though). So, it all boils down to a conflict with the page color.   Even though I'm using geometry's showframe option for "debugging" ends, and I don't mind to have no page color at that stage of development, is it possible to override this problem and have at the same time both the page layout displayed by geometry as well as the page colored?   Btw, should I open a new question given that the problem is actually different from the originally posted one?