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"Welcome to emergency mode!" Think it is a fsck problem Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete.  My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on the wrong track. Could you please help me?  edit: More from my attempt:  Suppose $A$ is a closed set and let $(x_n)$ be a sequence of points $A$ such that $\lim_n x_n\to b$.  Suppose now that $A$ has the property that $b\in A$, whenever $x_n$ converges to $b$. We know that every element of $x_n$ which is convergent in $X$ also converges to a point in $A$. Since $x_n$ is a Cauchy sequence in $A$, it must converge to a point $y\in A$. But the limit of a convergent sequence is unique.   Take $x\in A$ and select an appropriate $n$, which enables $x_n$ to converge to a point $x$ in $X$. Since the limit is unique, it must follow that $x=y$. Thus $x\in A$ and $A$ is closed.   If $A$ is a closed subset of $X$, then any Cauchy sequence of a point in $A$ is convergent in $X$ and hence converges to a point in $A$. Thus $A$ is complete. find a $\int_{x}^{g(x)} f(t)\,dt = 1$ Let $f : [0, 2] → \mathbb{R}$ be a continuous, positive function and  $$\int_{0}^1 f(x)\,dx = \int_{1}^2 f(x)\,dx= 1$$ For each $x \in [0, 1]$, prove that there exists a unique function $g(x) \in [1, 2]$ such that $$\int_{x}^{g(x)} f(t)\,dt = 1$$ and $g \in C^1$. Well, I think in $g(x) = x +1$, but I'm not sure. Any hint is appreciated.  Thank you. "ERROR: Extra }, or forgotten \right." in align* enviroment Why do I get a "ERROR: Extra }, or forgotten \right." in the following align* environment?  \begin{align*}     \int\left[\left(\der{E_z}{y}-\der{E_y}{z}-\mu\der{\alpha}{t}\right)\alpha&amp;+\left(\der{E_x}{z}-\der{E_x}{x}-\mu\der{\beta}{t}\right)\beta\\     &amp;+\left(\der{E_y}{x}-\der{E_x}{y}-\mu\der{\gamma}{t}\right)\gamma\left.\vphantom{\left(\der{E_y}{x}}\right]d\omega=0.     \end{align*}   thanks I have a string that represents a non indented XML that I would like to pretty-print. For example:  &lt;root&gt;&lt;node/&gt;&lt;/root&gt;   should become:  &lt;root&gt;   &lt;node/&gt; &lt;/root&gt;   Syntax highlighting is not a requirement. To tackle the problem I first transform the XML to add carriage returns and white spaces and then use a  tag to output the XML. To add new lines and white spaces I wrote the following function:   function formatXml(xml) {     var formatted = '';     var reg = /(&gt;)(&lt;)(\/*)/g;     xml = xml.replace(reg, '$1\r\n$2$3');     var pad = 0;     jQuery.each(xml.split('\r\n'), function(index, node) {         var indent = 0;         if (node.match( /.+&lt;\/\w[^&gt;]*&gt;$/ )) {             indent = 0;         } else if (node.match( /^&lt;\/\w/ )) {             if (pad != 0) {                 pad -= 1;             }         } else if (node.match( /^&lt;\w[^&gt;]*[^\/]&gt;.*$/ )) {             indent = 1;         } else {             indent = 0;         }          var padding = '';         for (var i = 0; i &lt; pad; i++) {             padding += '  ';         }          formatted += padding + node + '\r\n';         pad += indent;     });      return formatted; }   I then call the function like this:  jQuery('pre.formatted-xml').text(formatXml('&lt;root&gt;&lt;node1/&gt;&lt;/root&gt;'));   This works perfectly fine for me but while I was writing the previous function I thought that there must be a better way. So my question is do you know of any better way given an XML string to pretty-print it in an html page? Any javascript frameworks and/or plugins that could do the job are welcome. My only requirement is this to be done on the client side. Storing output of command in shell variable How to downgrade a package via apt-get? Colored borders around windows after update After some gorgeous updates I witness this: Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains? How can I change the umask for one command only? I know that in the case of Gaussian mixture, the "intuition" is that you're drawing from a PDF which itself is just a sum of weighted Gaussian PDFs.  I don't understand the intuition behind how the sum of two iid Gaussian RVs leads to a Gaussian RV. I know the maths behind it, but I'm having a hard time visualizing how the sum of two Gaussian RVs leads to a Gaussian RV (and not a mixture). Help evaluating a gamma function How do I upgrade from 12.10 to 13.04? Are +1/+1 counters different from +1/+1 effects? Is a +1/+1 counter I get from, say Unleash or Undying different than +1/+1 from something like  or ? How to calculate $f(x)$ in $f(f(x)) = e^x$? How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$?  Is there a faster-converging method than power series for fractional iteration/functional square roots? Is the vector cross product only defined for 3D? Wikipedia introduces the vector product for two vectors $\vec a$ and  $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and  $\vec b$, implying that $\vec a$ and  $\vec b$ are 3D vectors. Wikipedia mentions something about a 7D cross product, but I'm not going to pretend I understand that.  My idea, which remains unconfirmed with any source, is that a cross product can be thought of a vector which is orthogonal to all vectors which you are crossing.  If, and that's a big IF, this is right over all dimensions, we know that for a set of $n-1$ $n$-dimensional vectors, there exists a vector which is orthogonal to all of them.  The magnitude would have something to do with the area/volume/hypervolume/etc. made by the vectors we are crossing.  Am I right to guess that this multidimensional aspect of cross vectors exists or is that last part utter rubbish? Keep sharp edges when using subdivision surface How can I use the subdivision surface modifier to create more high poly meshes without ruining sharp edges? For example this mesh  The sphere should be subdivided but the rim of the cylinder should stay the same because it is a sharp edge. This is the result of the subdevision  How can I get sharp edges to not be affected by the sub surf mod? I am generating a fairly large index of .txt, .tex, and .pdf files. This process has worked for a long time, and the size of the index is not the issue. I have added logic to insert additional links in the index and now am running into:  TeX capacity exceeded, sorry [save size=80000]  Questions:  What are the typical causes of this kind of problem, and any suggestions on how to determine the cause of this. Can the save size be increased?  Of course any solution that just allows me to increase the save size would be preferable if it works, the first question is only important if the size can't be increased or I run into this problem again. Notes:  Generating smaller indices seems to work just fine, only when I attempt to generate the entire index do I run into problems, so coming up with a MWE will be difficult. Are you allowed to stack Divine Smite and Eldritch Smite on the same attack? Can I travel within the Schengen area if I currently don't have a valid visa or residency permit?