Prove: If S is countably infinite set, all finite subsets of S are also countably infinite
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
When he was just a professor, Dumbledore taught Transfiguration. As Headmaster what was he doing all day? In the 6th movie he was diligently looking for Horcruxes, but what about the other times?  When Fred and George pass the Maurader's Map to Harry they note that Dumbledore is pacing in his office. "He does that a lot" one of them says. Did Dumbledore just endless pace in his office as Headmaster, afraid to leave it for fear someone would know that he produced no work?   Sure, every year he had to go find a new DADA teacher, but most of those it seemed he did through correspondence. Horace Slughorn was the only one that seemed to resist. Dumbledore's 6th year may have been the only one where he earned his paycheck. So here is what i want to pull : mmcblk0 whith it's size 15.7 GB.  C:\adb&gt;adb shell shell@android:/ $ su su shell@android:/ # fdisk -l /dev/block/mmcblk0 fdisk -l /dev/block/mmcblk0  Disk /dev/block/mmcblk0: 15.7 GB, 15758000128 bytes 1 heads, 16 sectors/track, 1923584 cylinders Units = cylinders of 16 * 512 = 8192 bytes   This is what I tried:  C:\adb&gt;adb shell su busybox cat /dev/block/mmcblk0 &gt; C:\1.bin   I only have 1KB output with content inside said "unknown id". Also tried with pull but seems also doesn't work.  What should I do to get this 15.7 GB mmcblk0  ? How to pipe output from one process to another but only execute if the first has output? How can I rewrite this command to only email if there is output from the mailq | grep?  mailq | egrep 'rejected|refused' -A 5 -B 5 | mail -s 'dd' email@email   Is this even possible on one line?  See  for a more general case than sending email. Feature selection with Random Forests TexStudio IEEEtrantools package issue. User .cwl file? I am using TexStudio 2.9.4 on Windows 7. There is a problem that has been raised here at least once. It is that TexStudio flags unrecognised command  to  \begin{IEEEeqnarray}{rCl} and then within the environment flags expressions like \mathrm as being used outside the math environment. The fix (on Windows) is to manually edit the file C:\Users\USER\AppData\Roaming\texstudio\IEEEtrantools.cwl adding:  \begin{IEEEeqnarray}{cols}#\math,array \end{IEEEeqnarray} \begin{IEEEeqnarray*}{cols}#\math,array \end{IEEEeqnarray*}   in place of the \IEEEeqnarray entry there.   This works but is there a separate file that I could add (eg myfixes.cwl) which TexStudio would read? Burying the fix in a big autogenerated file is not ideal.  BTW is there any documentation on the format of .cwl files? I'm trying to prove that for $p,q&gt;0$, we have $$\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$    The hint given suggests that we express $\Gamma(p)\Gamma(q)$ as a double integral, then do a change of variables, but I've been unable thus far to express it as a double integral.  Can anyone get me started or suggest an alternate approach?  Note: This wasn't actually given to me as the $\Gamma$ function, just as a function $f$ satisfying $$f(p)=\int_0^\infty e^{-t}t^{p-1}\,dt$$ for all $p&gt;0$, but I recognized that. This is in the context of an advanced calculus practice exam. Why are my textures pink? Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$? If $a, b \in \mathbb{C}$, then we have the standard triangle inequality for the difference:  $$||a| - |b|| \le |a - b|.$$  I am wondering if this inequality generalizes to exponents greater that one.     My question is, for $1 &lt; p &lt; \infty$, does there exists a constant $C_p$ such that $||a|^p - |b|^p| \le C_p|a - b|^p$ for all $a, b \in \mathbb{C}$?   I am aware of the "standard" triangle inequality for $p &gt; 1$:  $$|a +b|^p \le 2^{p}(|a|^p + |b|^p),$$  and that $2^p$ may not be sharpest constant possible. If I try to use this estimate to prove, say, that $|a|^p - |b|^p \le C_p |a - b|^p$, I get stuck with an extra term that I'm not sure what to do with.  $$|a|^p - |b|^p \le 2^p|a -b|^p + (2^p -1)|b|^p.$$  A resolution on this matter is greatly appreciated. Why is everybody so concerned about etc/passwd? On iOS, while watching a video in YouTube app, when I go to the Home Screen, the video stops playing. I found no setting to enable it.  I know it is available for Android, but is there a way to do so in iOS? Is there any setting, or a 3rd party app that I can use? Distance between two hyperplanes I have two parallel hyper planes $$a^Tx=b_1,a^Tx=b_2$$ where $a \in \mathbb{R}^n, x \in \mathbb{R}^n ,b \in \mathbb{R}$ and I want to find the distance between the two. I have read that the distance between the two hyperplanes is also the distance between the two points $x_1$ and $x_2$ where the hyperplane intersects the line through the origin and parallel to the normal vector $\vec a$. These points are given by $$x_1=\frac{b_1}{\|a\|^2_2}a$$ and $$x_2=\frac{b_2}{\|a\|^2_2}a$$ Then the distance is $|x_1-x_2|$ but I don't really understand how we got $x_1$ and $x_2$. I am to prove by induction that given that $a+1/a$ is an integer (i.e. belongs to Z  ) then $a^n+1/a^n$ is an integer too. I'm pretty much clueless here. Thanks in advance. How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also? Can I be automatically notified of prolonged blocking in SQL server? About once a week I have to resolve a blocking chain on a SQL Server 2005 database, caused by a long-lived read lock from an Access 2003 front-end. The lock is taken out whenever a user opens a certain form and is released once the user has finished scrolling through the form or closes it. Since many of our users open this form as reference, these locks stay around for a while. Any update to the table causes the blocking, and suddenly nobody can select from this table since they're all waiting on the first lock. This is quite a problem for us, since lots of apps rely on this data. I understand that this locking behavior is part of how Access works with linked tables.  I've been solving the problem from Activity Monitor,  whenever I find out about it. This is a problem not only because it takes me time to do it manually, but also because it's reactive. By the time I hear about it, it's already been a problem for a lot of people.  I'd like to know if there is an automatic way to check for these long-lasting blocking chains, and either be emailed or have the problem resolved automatically. The logic seems straightforward enough ("if any process matching this SELECT query has been blocking for longer than a minute, notify me/kill it") but I don't know how to implement this with SQL Server.  For what it's worth, I think the proper solution is to fix or rewrite the app. However, due to departmental politics, this is not an option for the next few months, so I'm looking for a stopgap. How do I replace the *first instance* of a string in .NET? I want to replace the first occurrence in a given string.   How can I accomplish this in .NET? I learned recently that when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes. How does this alteration take place? Are there any best practices for choosing the title of web pages? There are a lot of resources online oriented toward SEO, but not toward users.  My impression is that most sites have something similar to Product - Page. But there are cases when switching this around is more useful. For example, if mail clients displayed the title like this:  Inbox (3) - Gmail  Instead of:  Gmail - Inbox (3)  You would be able to see the message count, even if the title was cut off. (I'm pretty sure Gmail used to do this).  I assume the title is mostly ignored by users except when:   using boorkmarks finding the right tab in the browser   What are the considerations for users when choosing a web page title? Are there different considerations for the home page, pages behind a login, etc? Is it ever bad to use \left and \right? For mathematics, is it ever disadvantageous to use \left(  and \right)  in equations?  I can't see any way that doing this would hinder the typesetting, so why isn't () just converted into \left( \right) automatically? This would save a lot of typing.  Cheers! Everybody who is using Lightroom probably has seen this. After importing RAW files, Lightroom shows a preview, but contrast and color of the preview change after a few short seconds to a slightly different view.  For some users the change is small, for others the change is huge. Colors are different (read: wrong), contrast is off.  Rumor says this is happening only (more?) for Nikon users.  What is Lightroom doing here?  I'd guess that for the first preview, LR is showing the JPG preview file, which is embedded in the RAW file. Then after a moment, LR renders the RAW file itself to show it's own interpretation.  But why is this interpretation so different from the embedded JPG? Static Classes In Java