Infinitesimal Values Does the $0.\overline{9} = 1$ proof show that the gap between numbers adjacent numbers is $0$, and so are infinitesimal values equal to $0$ ? Would $10^{-\infty}=0$ ?  Edit: I feel like the smallest non-zero number would be $0.0...1$, which when subtracted from $1$ would 'appear' to be $0.999...$ , but it can't since $1 = 0.999...$
Is it true that $0.999999999\ldots=1$? I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
Suppose $K/F$ is an extension field, and $E_1$ and $E_2$ are two intermediate subfields of finite degree. Let $E$ be the subfields of $K$ generated by $E_1$ and $E_2$. I'm trying to prove that $$ [E:F]\leq[E_1:F][E_2:F].$$  Since $E_1$ and $E_2$ are finite extensions, I suppose they have bases $\{a_1,\dots,a_n\}$ and $\{b_1,\dots,b_m\}$, respectively. If $E_1=F$ or $E_2=F$, then the inequality is actually equality, so I suppose both are proper extension fields. I think $E=F(a_1,\dots,a_n,b_1,\dots,b_m)$. Since $\{a_1,\dots,a_n,b_1,\dots,b_m\}$ is a spanning set for $E$ over $F$,  $$[E:F]\leq n+m\leq nm=[E_1:F][E_2:F]$$ since $m,n&gt;1$.  Is this sufficient? I'm weirded out since the problem did not ask to show $[E:F]\leq [E_1:F]+[E_2:F]$ which I feel will generally be a better upper bound. I want my intent to be launched when the user goes to a certain url: for example, the android market does this with  urls. so does youtube. I want mine to do that too. The usual Schrodinger Lagrangian is $$ \tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi, $$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real but differs from a real Lagrangian density  $$ \tag 2 \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $$ by a total derivative.  My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my Hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator Schrodinger equation.   Is it possible to work with the real Lagrangian density and somehow get the correct commutation relations? I would have expected two Lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two Lagrangians differing by total derivatives, how does one choose the correct one?   I guess the same thing happens for other first order systems like Dirac Lagrangian also. p:commandbutton action doesn't work inside p:dialog The strongest player in the game is a close ally of mine and we each founded and follow of one of the strongest two religions in the game, with little cross-over between religions within our territories.  I've just spotted that he has a Great Prophet approaching my territory. According to the Civopedia they are powerful at spreading religion, and I'd rather not have a huge hit to the followers of my religion in my cities.  How can I prevent another players Great Prophet from damaging the religion in my cities, preferably without causing a major diplomatic incident. The 1st law of thermodynamics If a gas expands adiabatically will the work done be positive or negative?  I think it will be positive as $\delta W=p dV$ and we have a positive sign due to the work done by the gas is positive as it pushes to expand. But this site has the negative of the answer I obtain.   (See adiabatic processes section) When should you use a class vs a struct in C++? How can I get out of the debt-trap as quickly as possible? I have a credit card with 30% interest and a real estate loan with 11% interest. The amount of debt on my card is lower than the real estate loan.   I want to redeem the installments of the real estate loan, but I can't do it because the credit card debt takes all my money.    What can I do in this situation? How can I get out of the debt-trap as quickly as possible?   I have a 4-room house and 2 cars, and 2 children who have to go to school, that's why there are a lot of fix expenses. When punctuating the dimensions of something, if both dimensions share the same units, does the punctuation occur after each measurement or just after the last one. For example, if I'm looking at a sheet of paper, which is correct: {8 1/2 x 11"} or is {8 1/2" x 11"}? From Geneva airport to Lausanne station On September 4, I will be arriving in Geneva by airplane at approximately 11 AM. After that, I need to go to Lausanne. I have two questions:   Do I need to buy a train ticket online, or can I buy it directly at the station without any problems? (Problems such as: the trains are always full, it's not permitted to buy a ticket etc.) Moreover, are the tickets only for a specific train or it is like "you can board any train from Geneve to Lausanne with this ticket"? How do I get from the airport to the Geneva train station (as low a budget as possible, I guess walking is not an option though), and how much does it cost?   I'd appreciate answers from people who have been there, but if you're able to find an exhaustive link from any website that's okay too. Remove completed reminders in OS X and iOS Local DNS server Geometric meaning of the determinant of a matrix What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.  Thank you very much. When I visit a site I haven't visited (while logged in) for a while, I see:  Welcome back! If you found this question useful, don't forget to vote both the question and the answers up.  That's often intended behavior. But today I visited  and was shown that banner even though the question and all its answers are locked. Voting arrows aren't even shown on any of the posts on that page. I believe that banner should be disabled on pages where there is no post that can be voted on by anyone. Travelling with two different passports Outlook Rule - Move mail that is READ and Older than X days Outlook 2010. Want to create a rule that moves all mail from my inbox to another folder:   Has been read Is older than X days   I was looking at Auto-archiving, but it does not seem to let me be this specific with my criteria. Restore All Databases Script I am migrating SQL Server DBs to a new instance.  I have been told that it is possible to dynamically build a RESTORE script from available backups in the system catalog.  Does anyone know of a sample script to do this?  Thanks! For context, see .  In essence:     "This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation (not including the ) on this site."   Suggested addition to existing message is in bold. I have made a script that will sort files according to their extension and place them in the proper folder. For example, place abc.jpg in the directory jpg.  #!/bin/bash #this script sorts files according to their extensions oldIFS=$IFS IFS=$'\n' (find . -type f) &gt; /tmp/temp for var in `cat /tmp/temp` do name=`basename "$var"` ext=`echo $name | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2- | cut -d'.' -f2-` mkdir -p $ext mv "$var" $ext/ 2&gt; /dev/null done IFS=$oldIFS   problem with this script:    it involves use of IFS, it is said to avoid use of IFS, as much as possible it does not sorts file without file extensions  it will sort files like abc.tar.bz in folder named bz, but however such a file should go in tar.bz folder see line 9 of my script; if any file contain more no. of dots(in its name) than no. of cut -d'.' -f2- in the script than if will result in file name taken in extension part. for example, a file named i.am.live.in.india.and.i.study.computer.science.txt will be placed in folder named study.computer.science.txt   you may also suggest any tweaks to make this script more smaller and neat. What to do with bounty on question which is no longer relevant?