If $\{a_n\}$ is a convergent sequence of real numbers, then $ \lim_{n \to \infty} n(a_{n+1} - a_n) = 0$
Proving that the difference of the terms of a sequence, multiplied by $n$ converge
I need to view the saved WiFi passwords on my non-rooted Android phone I have an LG Optimus 7 (not-rooted), and all my WiFi passwords are there. I want to view the saved WiFi passwords to transfer thm to my new Samsung Galaxy S3, but I don't know how.  I tried to search on the Internet and Google play for an application that would allow me to view this information, but I can't find one. Is there any way for me to accomplish this? Relativity and Galaxy Rotation Speed Don't heavier objects actually fall faster because they exert their own gravity? The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half. The halves clearly can't then fall more slowly just by being sliced in two.  However, I believe the answer is that when two objects fall together, attached or not, they do "fall" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $$ F_{\text{g}} = \frac{G m_1 m_2}{d^2} $$  Given $F = ma$ thus $a = F/m$, we note that the mass of the small object doesn't seem to matter as when calculating acceleration the force is divided by the $m$ term, its mass. However, this overlooks that the force is actually applied to both objects, not just to the smaller one. The acceleration on the second, larger object is found by dividing $F$, in turn, by the larger object's mass. The two objects' acceleration vectors are exactly opposite, so closing acceleration is the sum of the two:  $$ a_{\text{closing}} = \frac{F}{m_1} + \frac{F}{m_2} $$  Since the Earth is extremely massive compared to everyday objects, the acceleration imparted on the object by the Earth will radically dominate the equation. As the Earth is $\sim 5.972 \times {10}^{24} \, \mathrm{kg} ,$ a falling object of $5.972 \, \mathrm{kg}$ (just over 13 pounds) would accelerate the Earth about $\frac{1}{{10}^{24}}$ as much, which is one part in a trillion trillion.  Of course in everyday situations, we can for all practical purposes treat objects as falling at the same rate because of this negligible difference—which our instruments probably couldn't even detect. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening.  Am I right or wrong?  What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Moon. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and thus "falling" consists of multiple objects mutually attracting in space. Why doesn't calling a Python string method do anything unless you assign its output? If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$ Why is the root directory denoted by a / sign? I have done some research about this on Google, but the results were cloudy. Why is the / sign used to denote the root directory. Are there any solid reasons behind it? How accurate is the GPS on the iPhone's Find My iPhone app?  I used Find My iPhone app to track a friend. The first 3 stops were dead on but when it refreshed it showed them to be several streets over from actual location. What could cause that? Derivation of Frey equation from FLT I understand, on  a layman's level, Fray's motivation to write an elliptic equation corresponding to an assumed solution to FLT. My question is, how technically is Frey's equation derived?   $1.$ FLT : Contradiction. There exist integers $A,B,C,n&gt;2$ such that $A^n + B^n = C^n$  $2.$  Frey: $y^2 = x(x-A^n)(x+B^n)$  $3.$  Elliptic curve $y^2= x^3 + ax + b$    $4.$  The curve is not modular. => not elliptic => FLT  proved.   I need is the algebra to go from 1. to 3. above.  Assume Numbers A,B,C such that A^n + B^n = C^n contradicting FLT.  $y^2 = x^3 + ax + b$ an elliptic curve in Weirstrass form. I am guessing A and B go unchanged into x(x-A^n)(x+B^n)   Is that correct ?  Expanding 2. I get a term in x^2 , there is none in 3. This is my problem.  What does a $2$ dimensional graph of the Frey curve look like ? How to completely disable auto-save and versions in Mac OS X Lion? I hate the automatic saving and versioning in Mac OS X Lion.  Is there a way to completely turn it off?  (This has been asked before , but the answer and the linked article are completely wrong, so I thought I'd try again.) I am getting a NoClassDefFoundError when I run my Java application.  What is typically the cause of this? Recently I paid a lot (almost one month of my salary) for repairing my car. I am wondering if there is a way to get deduction on federal or state taxes (USA, California). Two "offtopic" flag reasons Updating list definition using features I have a feature that deploys content types, a list definition and template that uses these content types and finally a list instance. This feature is already in production.  Now we have a change request to add new views to this existing list. So I make the changes to schema.xml and upgrade the solution. But alas, the new views don't show up. If I create a new site, the list instance is created with the new views. I tried deactivating/reactivating, iisreset but still no success. Is there anything I'm missing? How do different difficulties work on story missions? When selecting a story mission, I can choose between "Easy" and "Hard".  Next to the difficulty, the game displays some kind of level.  However, when you start the mission, you still end up in the common map area with other players.  What exactly changes when you select a higher difficulty?  Is it possible to see other players playing on "Easy" when you select "Hard" before entering the mission zone where respawning is restricted? Largest Mass Diffraction Difference between Unity and GNOME I often come across the terms Unity and GNOME while reading about Ubuntu. I understand that Unity is the default desktop environment for Ubuntu. What is GNOME and how is it related to/different from Unity? Since I can't find any alternative to Linux'  elevation command, I have the following question:  How to define a PowerShell function which requires elevation? I mean UAC prompt.  Say, such function follows:  function system-check {     SFC /ScanNow }   System:  Windows 8.1 Pro 64-bit  PowerShell:   Major  Minor  Build  Revision -----  -----  -----  -------- 5      0      10586  117    EDIT1:  To be 100% understandable let me rephrase:   I run PowerShell as user I run the aforementioned function system-check I want the function to elevate in order to be able to execute the command; note, that I want the UAC prompt to appear I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty hypersurface, then $X$ and $H$ meet.   Following the hint, I've done the following: By assumption, $X$ is given by $V(I)$ for some homogenous prime ideal $I\subset k[x_0,\dots x_n]=:A$ and $H$ by $V(f)$ for some homogenous polynomial $f\in A$. We wish to prove that there is a homogenous prime ideal $\mathfrak{p}$ (that's not the irrelevant one) containing both $I$ and $f$.  Consider the affine cone $\mathrm{Spec}\ A/I$ of $X$ in $\mathrm{Spec}\ A$. Being homogenous, both $I$ and $f$ are contained in the ideal $(x_0,\dots x_n)$. By Krull's principal ideal theorem, every minimal prime ideal containing $f\mod I$ has codimension one. $X$ having dimension at least one implies that there are homogenous prime prime ideals $\mathfrak{q}\supsetneq \mathfrak{r}\supset I$. This implies that $(x_o,\dots x_n)$ has codimension at least two. Thus there are prime ideals containing $f\mod I$ that are contained in $(x_o,\dots x_n)$. However, I'd need to show that there are homogenous such prime ideals, right? I'd rather get hints than full solutions. Default constructor with empty brackets Align multiple numbered aligned blocks There are many discussions over how to align multiple align environments with interspersed texts. However, I want to label multiple aligned blocks such that each number will center vertically for each aligned blocks and at the same time align across blocks.  Here is a MWE that achieves either at one time. The first part simply uses one giant align environment with some \nonumber directive. The second one use equation to wrap aligned but lose alignment info. How should I achieve the best of both worlds?  \documentclass{article} \usepackage{amsmath} \begin{document}   % Part I   \begin{align}           x^2 + y^2 &amp;= 1 \\                 z^2 &amp;= 0 \nonumber \\     x^2 + y^2 + z^2 &amp;= 1   \end{align}    % Part II   \begin{equation}     \begin{aligned}       x^2 + y^2 &amp;= 1 \\             z^2 &amp;= 0 \\     \end{aligned}   \end{equation}   \begin{equation}     \begin{aligned}       x^2 + y^2 + z^2 &amp;= 1     \end{aligned}   \end{equation} \end{document}