The question referred from this discussion   I understand the proof of the Leems.But i still don't get "Then almost all x ∈ R belong to at most finitely many of the E_Ks"? i understood that if x belongs to infinitely E_k then the set contain all such x has measure 0 but how it implies finitely many x's belongs to that set? or infinitely many x ∈ R belong to at most finitely many of the Ek's? how i can get a contradiction?  THanks in advance for the help.
The Borel-Cantelli Lemma in  seems to be a sort of "corollary" of the .  It says:  "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. Then almost all x $\epsilon \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"  I don't get three things:   "Then almost all x $\in \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"   What does that mean? Since the Borel-Cantelli Lemma comes after the "almost everywhere" definition, I guess: $\exists \ Z \ \subset \mathbb{R}$ s.t. $m(Z) = 0$ and $\forall x \ \in \ \mathbb{R} \setminus Z$, x belongs to $(E_{j}, E_{j+1}, E_{l}, ..., E_{m})$.   In the proof, one can see the $m(\limsup \ E_k) = 0$ part, stated as $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$. For some reason, $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$ implies that "Therefore almost all x $\in \ \mathbb{R}$ fail to belong to $\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]$ and therefore belong to at most finitely many $E_k$'s"   Okay, why? What is the contradiction if almost all $x \epsilon \ \mathbb{R}$ belong to all the $E_k$'s? Or a countably infinite subcollection?   What is the significance of this? Why not just state the Borel-Cantelli Lemma as "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$"?
Does a shortcode with a single attribute have to use an array? Defrosting meat/fish in warm water I have found warm water to be a great way to quickly defrost meat and fish as opposed to just letting it sit outside the freezer, which is what my mom does.  But I wonder if there are any negative side effects to the method, as in, for example, compromised taste or compromised food safety.  I do understand that the water shouldn't be too hot as to avoid actually cooking the frozen food. What are these  error below ?  Package natbib Warning: Citation `Krishna2014' on page 16 undefined on input li ne 502.  [16 &lt;./figure/aks smr.png&gt;] [17] [18] ! Undefined control sequence. l.560 \Bibliography                    {figure/bib} Why does a helium filled balloon move forward in a car when the car is accelerating? What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$? Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find a primary decomposition of $I$.    I tried to draw the graph of the variety of $I$ and get a decomposition of $(x,y)\cap(x,z)$ (the two prime ideals corresponds to the irreducible components of the variety), but apparently these two ideals are not the primary decomposition of $I$, and I don't know how to fix this.  I would really appreciate it if someone can help me. thx Calculating the expected values of the min/max of 2 random variables Is it possible to change the default width of an image included via \includegraphics (from the graphicx package) to a custom value, for example \linewidth.  So I want something like this:  \documentclass{article}  \usepackage[draft,width=\linewidth]{graphicx}  %or something like \setglobalimagewidth=\linewidth  \begin{document}  \includegraphics{test1} %width=\linewidth \includegraphics[width=4cm]{test1} %width=4cm   \end{document} 16.04 LTS upgrade alert- How to handle it? I am running Ubuntu 14.04 LTS. Today when I logged into my computer I got an update alert saying that I could upgrade to Ubuntu 16.04 LTS. The other options were not now or do not upgrade. If I chose to upgrade will it delete my /home directories and the installed software under /opt and /usr/local ? Is a music band a singular or a collective entity, grammatically speaking? Duplicate:    What is correct to say?  Korn* is a great band  OR  Korn* are a great band.  (* You can replace your favourite band's name here) Of course everybody there is no doubt about the following sentences:  The Beatles are a great band Led Zeppelin is a great band  But with certain kinds of names, the situation becomes difficult.  Metallica are a great band  OR  Metallica is a great band  Is there any consensus on this matter? Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$ I'm having trouble with proving the following for my math study:  Let $x$ be a real number with $|x| &lt; 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$  I tried this:  $\sum_{n = 1}^\infty n^qx^n$ = $x + 2^q + 3^qx^3 + ...$ $= x(1 + 2^qx + 3^qx^2 + ...)$  So far, I don't know how I have to do this, I tried to apply the root test on this, but for that I would need to compute the lim sup of the series, so I thought that was not correct. Could you explain me how to prove this?  Thanks in advance! Now that El Capitan is "rootless", is there any way to get dtrace working? I've found dtrace to be an invaluable tool for debugging and troubleshooting all sorts of issues, not to mention the two dozen dtrace toolkit scripts shipped by Apple as part of El Capitan.  On El Cap though, running dtrace usually results in an endless spew of errors making dtrace all but useless. How to look up a symbol or identify a letter from a math alphabet or other character? I know what my symbol or character looks like, but I don't know what the command is or which math alphabet it came from. How do I go about finding this out? How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? Ways to put numbered balls in boxes no box being empty I know the formula for putting $n$ identical balls in $r$ different boxes such that each box has at least 1 ball, but what is the formula for putting $n$ different balls in $r$ different boxes, no box being empty?  Thanks! How do I join two separate objects so I can edit them as one mesh? If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$  Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$  b) show that $ d$ and $ d_b $ are equivalent metrics. c) let $(X,d) $ be $(\mathbb{R}, |\cdot|)$ Show that there exists no $ M&gt;0$ such that $ |x-y| $ $\leq$ $Md_b$ $(x,y)$ for all $ x,y$ $\in$ $\mathbb{R}$  I have calculated $f(t)$ and $f'(t)$ and from this I know $f(t)$ is an increasing function as $f'(t)$ is strictly positive. But I don't know where to go from here or how to do parts b) and c) question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$ Suppose I installed ubuntu with the user usr1 then I added usr2. I then create a group grp1 where this 2 users are added. A folder is owned by usr1 and has group grp1. But when I login as usr2 I cannot write. The folder has permissions 755...  usr2@kmserver:/media/data/www$ groups usr1 usr1 : grp1 ... usr2@kmserver:/media/data/www$ groups usr2 usr2 : grp1 ... usr2@kmserver:/media/data$ ls -l total 20 drwx------ 2 root root 16384 2011-05-12 10:53 lost+found drwxrwxr-x 2 grp1   grp1    4096 2011-05-12 15:15 www   Whats wrong? How do you Programmatically Download a Webpage in Java I would like to be able to fetch a web page's html and save it to a String, so I can do some processing on it.  Also, how could I handle various types of compression.  How would I go about doing that using Java? I cannot see WiFi option in my Ubuntu 15.04. What should I do? Revert asap  lspci -nnk | grep 0280 -A2 output:-  03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30) Subsystem: Foxconn International, Inc. Device [105b:e09a]   rfkill list output:-  0: hci0: Bluetooth Soft blocked: no Hard blocked: no 1: acer-wireless: Wireless LAN Soft blocked: no Hard blocked: no      dmesg | grep ath [    9.545199] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [   10.029684] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [   10.029701] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-5.bin failed with error -2 [   10.029704] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-5.bin': -2 [   10.029711] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-4.bin failed with error -2 [   10.029713] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-4.bin': -2 [   10.029719] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-3.bin failed with error -2 [   10.029721] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-3.bin': -2 [   10.029726] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-2.bin failed with error -2 [   10.029728] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-2.bin': -2 [   10.029733] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware.bin failed with error -2 [   10.029735] ath10k_pci 0000:03:00.0: could not fetch firmware (-2) [   10.029737] ath10k_pci 0000:03:00.0: could not fetch firmware files (-2) [   10.029739] ath10k_pci 0000:03:00.0: could not probe fw (-2)