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How to install the latest versions of NodeJS and NPM? I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot! What graphics packages are there for creating graphics in LaTeX documents? What packages do you use and recommend for creating graphics in your LaTeX documents?     As this is a community wiki post, please add your package to the accepted answer (or add a comment, and someone with >100 rep will add it to the CW answer), and include a brief description of what differentiates it from others and how it can be used (GUI drawing tool which generates code, type in raw text, or generates image for inclusion in document).   We'll eventually sort these answers under headings. How do I embed a gather environment within a custom environment? I've tried defining the following custom environment, but I get errors:     \begin{gather} ended by \end{solution}   etc...   Here's a MWE showing my problem:  \documentclass{article}  \usepackage[fleqn]{mathtools}  \newenvironment{solution}{%  &lt;-Solution   \par\addvspace{1em}   \itshape{Solution:}\par\begin{gather}}   {\end{gather}\par\normalfont\hfill\qed\par\addvspace{\medskipamount}}%  \begin{document}   \begin{gather}     testing   \end{gather}    %% This does not work:   \begin{solution}     testing   \end{solution} \end{document} Proving the measure of an increasing sequence of measurable sets is the limit of the measures Show that if $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ is an increasing sequence of measurable sets (so $A_j\subseteq A_{j+1}$ for every positive integer $j$), then we have $$m\left(\bigcup_{j=1}^\infty A_j\right)=\lim_{j\to\infty}m(A_j)$$  Here is my proof: According to the $\sigma$-algebra property, $\bigcup_{j=1}^{\infty}A_j$ is a measurable set, so it makes sense to talk about $m(\bigcup_{j=1}^{\infty}A_j)$. Firstly, I prove that $\lim_{j\to\infty}m(A_j)\leq m(\bigcup_{j=1}^{\infty}A_j)$. This is because for any given positive integer $N$, $A_N\subseteq \bigcup_{j=1}^{\infty}A_i$, according to monotonicity, we have $m(A_N)\leq m(\bigcup_{j=1}^{\infty}A_i)$. Take the limit,we will have $\lim_{j\to\infty}m(A_j)\leq m(\bigcup_{j=1}^{\infty}A_j)$. Secondly, I prove that $m(\bigcup_{j=1}^{\infty}A_j)\leq \lim_{j\to\infty}m(A_j)$. For any given positive integer $N$, $\bigcup_{j=1}^N A_j = A_N$. According to monotonicity,we have $m\left(\bigcup_{j=1}^N A_j\right)=m(A_N)\leq \lim_{j\to\infty}m(A_j)$. Take the limit, we will have $m\left(\bigcup_{j=1}^\infty A_j\right) \leq \lim_{j\to\infty} m(A_j)$. Combine the above two arguments, we will see that $$m\left(\bigcup_{j=1}^\infty A_j\right)=\lim_{j\to\infty} m(A_j)$$$\Box$  The above is my proof, unlike many books, my proof does not use the property of countable additivity. So I doubt my proof is correct. Who can point out where are my mistakes? GameJolt is a gaming platform/website that offers API access with features like achievements tracking for users, etc. I can use the API no problem on my tests but what I'm concerned about is the private key that is issued to my game. How do I ensure that it's protected from people that want to decompile my game? Once  they get the API private key, they will be able to do anything with it. Using the same variable as a fixed and random effect in Mixed Effect Models Lyman and Balmer series Captchas not displaying in SO for me When I get a captcha on Stack Overflow, I don't see the image or the input box with Firefox. I originally had the "HTTPS Anywhere" extension installed but still have the same issue when it's disabled (This caused issues previously when Google's SSL certs were expired for an old domain IIRC).  The requests to Google seem to be never-ending; browsing to the URLs that are still loading results in some pages of JavaScript, but no errors as you'd hope.  The screenshot was taken after the page had been open for more than 3 minutes, and I'm not sure why the GETs are still running. The request popup visible in Firebug is for "GET challenge?k=...."  When browsing to the same captcha page in IE, there are no issues.  I'm aware this could be an issue with my Firefox configuration, as opposed to the site, but would appreciate it if someone could verify if it's just me, and suggest a fix.  EDIT   While visiting the Recaptcha site, I can see a Captcha with no problems If I right-click the problem URLs in FF and Open in new tab, the pages load fine. From the fact that FB omits IP Addresses for the problem ones, I assume that it's not even opening a connection (nor perhaps performing a DNS lookup) Other sites that use Recaptcha seem to work fine (I picked 3 at random from Google) Exclude bounties from affecting privileges So I have a torus, and I need it to "telescope" open -- kind of like Pacman's death animation, but with a torus instead of a sphere.  I'm very new to 3D modeling/animating so I really have no clue how to tackle this problem.  Any advice to help me get pointed in the right direction would be most appreciated :) How to hide TabPage from TabControl How to hide TabPage from TabControl in WinForms 2.0? Lubuntu 16.10: How to make login through greeter without typing password work Projection on the hyperplane $H: \sum x_i=0$ Consider the hyperplane $H: \sum_{i=1}^n x_i=0$. Let $P$ denote the orthogonal projection onto $H$. What exactly is the matrix $P$ and how do we find it? Given two equivalent norms, convergence in one implies convergences in other. Suppose we have two equivalent norms, $\|\cdot \| \sim \left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert$. We assume that $x_n \rightarrow x $ in $(X,\|\cdot \|)$. Show that $x_n \rightarrow x $ in $(X,\left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert)$  We observe since the norms are equivalent, there exits constants $\alpha, \beta$, such that $\alpha \|\cdot \| \leq \left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert \leq \beta \|\cdot \|$, with $0 \lt \alpha \leq \beta$  Since, $x_n \rightarrow x$ in $(X,\|\cdot \|$) we know that for all $\epsilon&gt;0, \exists K(\epsilon)$ such that for n>$K(\epsilon),$ we have $\|x_n -x\| \leq \epsilon$  Since, we know $\beta \gt 0$ , then $ \beta \|x_n -x\| \leq \beta \epsilon$, Thus we see that $\left\vert\!\left\vert\!\left\vert x_n-x\right\vert\!\right\vert\!\right\vert\leq \beta \epsilon$.  This is how far I've gotten, our goal to is to find a a $J(\epsilon)$ such that for all $\epsilon&gt;0, \exists J(\epsilon)$ such that for n>$J(\epsilon),$ we have $\left\vert\!\left\vert\!\left\vert x_n-x \right\vert\!\right\vert\!\right\vert \leq \epsilon$  My candidate for $J(\epsilon$) is $J(\epsilon) = \frac{K(\epsilon)}{\beta}$ from my above argument, but I'm not exactly sure how to validate this. If someone could help me on the proof, it would be very appreciated. I believe I'm pretty close, just missing something small. How can I scale all images in a folder to the same width? I want to scale all images in a given folder to the same width (but different appropriately scaled heights). How can I do this using  a GUI-based or command-line tool?  For bonus points, is it possible to restrict which images are scaled based on their initial width (that is, only scale images that have a width > x and/or a width &lt; y)? Can "mustn't" be used for conclusions? I heard this sentence in an American film a while ago as I was watching it on DVD (the part after but is verbatim):  "I'm doing my best but I mustn't be doing it right."  This is something I occasionally hear in American films: phrases like "he mustn't have done it" or "she mustn't be studying now", where a logical conclusion is expressed. So far I've thought that the normal thing to say is "he can't have done it" or "she can't be studying now".  My question is: Can mustn't be used to express a logical conclusion when the speaker is certain that something didn't happen or isn't happening, at least in informal speech? Is this an American usage? (I've never heard this usage in British English, but this doesn't mean it doesn't exist.) Is there a change going on in the usage of the particular modal verb?  Note: All the references I've checked don't even mention this use of mustn't. Google books aren't of help either.   EDIT: I should clarify that I'm asking this question because if I wanted to express that I'm certain I'm not doing something right (as in the sentence quoted from the film) I'd say "I'm doing my best but I can't be doing it right". I would think that the use of mustn't/must not wouldn't be standard usage (although the meaning is perfectly clear to me; I never mistook it for an injuction). Cartographic techniques for symbolizing routing data Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. Why do I have to click twice to submit a question when Stack Exchange's automated question review has no advice to show? Where can I obtain an English dictionary with structured data?