Let $A_n$ be a sequence of independent events with $$lim_{n \to \infty}P(A_n)=0.$$ Suppose that  $$\sum_{n=1}^{\infty}P(A_n^c \cap A_{n+1}) &lt; \infty.$$ Will the events $A_n$ cease to occur eventually with probability one?  I think that the question is asking to show that  $$P(A_n\ i.o.)=0$$ And I tried to prove that $$\sum_{n=1}^{\infty}P(A_n)&lt;\infty$$ So that I can use the Borel-Cantelli Lemma.  But I did not get nowhere. I am kind of confused about all these operations includes limits, sets, and interplays between them, as well as the $limsup$ and $liminf$ stuff. Hope someone could help me.
Practice problem for exam:     Let ${A_n}$ satisfy $\sum\limits_{n=1}^\infty P(A_n \cap A^c_{n+1}) &lt; \infty$ and $\lim\limits_{n\to \infty} P(A_n) = 0$.  Show that $P(\lim \sup A_n) = 0$.    I can see that it is sufficient to show that $P\left(\sum\limits_{n=1}^\infty A_n\right) &lt; \infty$ and then apply the Borel-Cantelli lemma, but I am having trouble showing this from the information given.  I see an almost identical problem here: , however the hint there is not enough for me to make headway, and this problem is also different in which term gets the complement inside the first sum.
How can I animate the Restrict Visibility property of an object? I've been trying to create an animation where an object blinks on and off.   For example, I have a Cube and I set a keyframe on it's Restrict Visibility icon (the "eye" icon) with I, then I toggle off that "eye" so my cube doesn't show in the Viewport and again I press I to record that keyframe. Then, when I preview the animation in the Viewport, I see my cube appears and then disappears like I want, but when I render it, the cube does not disappear at all.  What am I doing wrong? How to use Terminal to delete all .svn folders recursively? What command can I type into the Terminal so that I can delete all .svn folders within a folder (and from all subdirectories) but not delete anything else? Quick way to subtract white outline from knotwork pattern in Illustrator? I am working with a pretty complex traced AI that I received, and I want to basically make it so that all the visually black stuff is shapes, and all the white outlines (invisible here) are removed. Most of the spaces when a line crosses over another is created by a white outline.   I know about expanding appearance and the pathfinder tool, but because there are so many lines that are crisscrossing, I haven't had an easy way of minusing the fronts away. I could expand appearance and sort every object carefully, but I feel like there may be a quicker way of doing this.  If I put it in Photoshop as a smart object in 'screen' mode, it looks right.. I could put that back in Illy and trace it, but I don't want to complicate the paths. Can I apply multiple background colors with CSS3? Can't install php5 on Ubuntu 16.04 so I've recently installed my first Ubuntu, to be precise, a daily build of 16.04, on my new laptop since the hardware was to new for 15.10. Now I'm in the process of setting it up and I want to install php5 and some extensions.  When I execute sudo apt-get install libapache2-mod-php5 I get the error message that there is no installation package available for php5 and libapache2.  The error message is in german, so I doubt that this would help, but anyways here it is. :-) E: For package »libapache2-mod-php5« existiert kein Installationskandidat.  Do I need to add a apt-get repository? Or what am I doing wrong? If you look at this product:    you can clearly see that it is a DVD+R, i.e. once it is written, it cannot be erased/rewritten.  Then what does the "RW" logo (in the lower right corner) mean? I thought it stood for ReWritable, but experimenting with the medium shows that it is not. Rooting Stock Firmware - GSII with NFC $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$ I'm trying to understand why $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$.  Let $\dots\to F_1\to F_0\to M\to 0$ be a free resolution of $M$. Then $$ F_1\to F_0\to M\to 0$$ is exact. Hence $$ F_1\otimes_R N\to F_0\otimes_R N\to M\otimes_R N\to 0$$ is exact. Call the left map $f_1$ and the right map $f_0$. Then $M\otimes N\simeq F_0\otimes N/\ker (f_0)\simeq F_0\otimes N/\text{im}(f_1)$.  But if I understand the definition correctly, to show the result we need that $M\otimes N\simeq H_0(\mathcal F)=\ker f_0/\text{im}(f_1)$. But why does $\ker f_0=F_0\otimes N$? If it does at all... Why do I get "Required key not available" when install 3rd party kernel modules or after a kernel upgrade? Can a continuous real function take each value exactly 3 times? I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. Book about space alien cats vertical alignment of text in tikz node The output from the following script \documentclass{article} \usepackage{tikz} \usepackage{graphicx} \begin{document} \begin{tikzpicture}   \node(title)[text width = 0.25\textwidth, text height = 3 cm, draw]{title};   \end{tikzpicture} \end{document}  is  How can I vertically align the text in the node? Specifying the text depth as in \node(title)[text width = 0.25\textwidth, text height = 1cm, text depth = 1 cm, draw]{     title line 1\\     title line 2\\     title line 3};  Does not result in the text being vertically centred. Problems when adding a fillable PDF form if the page is rotated How can I change GNOME Shell theme? So I'm trying to power 2 of these  together in series. I know that I have to add the voltage of the two lights together and get a resistor that restricts the power to that level. I have a 9V battery and the lights are 3V each. I brand new to all of this, and understand almost nothing, so any help would be appreciated. Can someone help me choose the correct resistor (And how to identify that resistor with the color branding thing) so two of those lights listed above can be wired in series? Thanks! First, I'll state some background that lead me to the question.  I was thinking about quantization of space-time on and off for a long time but I never really looked into it any deeper (mainly because I am not yet quite fluent in string theory). But the recent discussion about  got me thinking about the topic again. Combined with the fact that it has been more or less decided that , I decided I should give it a go.    So, first some of my (certainly very naive) thoughts.  It's no problem to quantize gravitational waves on a curved background. They don't differ much from any other particles we know. But what if we want the background itself to change in response to movement of the matter and quantize these processes? Then I would imagine that space-time itself is built from tiny particles (call them space-timeons) that interact by means of exchanging gravitons. I drawed this conclusion from an analogy with how solid matter is built from atoms in a lattice and interact by means of exchanging phonons.  Now, I am aware that the above picture is completely naive but I think it must in some sense also be correct. That is, if there exists any reasonable notion of quantum gravity, it has to look something like that (at least on the level of QFT).  So, having come this far I decided I should not stop. So let's move one step further and assume string theory is correct description of the nature. Then all the particles above are actually strings. So I decided that space-time must probably arise as condensation of huge number of strings. Now does this make any sense at all? To make it more precise (and also to ask something in case it doesn't make a sense), I have two questions:        In what way does the classical space-time arise as a limit in the string theory? Try to give a clear, conceptual description of this process. I don't mind some equations, but I want mainly ideas.   Is there a good introduction into this subject? If so, what is it? If not, where else can I learn about this stuff?      Regarding 2., I would prefer something not too advanced. But to give you an idea of what kind of literature might be appropriate for my level: I already now something about classical strings, how to quantize them (in different quantization schemes) and some bits about the role of CFT on the worldsheets. Also I have a qualitative overview of different types of string theories and also a little quantitative knowledge about moduli spaces of Calabi-Yau manifolds and their properties. Negative probabilities in quantum physics If we set $A$ and $B$. Assuming that there exist an injective map from $A$ to $B$ and an injective map from $B$ to $A$. How to prove that there is a bijection between $A$ and $B$. I have data in an Excel file that I want to join with a shapefile in my project. The Excel file contains values like volumes. The Excel file and the shapefile I want to join it with both contain the same property names.  The problem is, when I join the two, I get too many decimals in the attribute table. I get six, but I only want it to be one.  I tested to change the format of the cells to one decimal, and then I saved the file as *csv. But, somehow I seem to loose data in the process? It should be 42 rows that contains data in the column i am working with, but I end up with only like 10?  Another thing is that my property names contains the letters Å. Ä and Ö, perhaps this complicates things?  I read the question , but it doesn't mention the problem with the data loss.  My question was tagged with , but I didn't write what version I'm using. I use QGIS 2.2.0. Would it help to upgrade to the latest version?