Let's suppose that $a_n$ satisfy $|a_{n+1}-a_n|&lt;2^{-n}$ for all $n$ . How can I prove that $a_n$ is a cauchy sequence?
Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy.   My Work  So by the definition of a Cauchy sequence, for all $\varepsilon &gt; 0$ $\exists N$ so that for $n,m \ge N$ we have $|a_m - a_n| \le \varepsilon$. However, questions like  make me understand that the $2^{-n}$ condition is necessary for this to be a true statement.   So I am wondering how to appeal to the Cauchy definition for this proof. Do I prove that every convergent sequence is therefore Cauchy, and then try to prove convergence?
Deriving the Lagrangian for a free particle I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading &quot;Mechanics&quot; by Landau and Lifshitz recently and hit a few roadblocks right away.  Proving that a free particle moves with a constant velocity in an inertial frame of reference ($\S$3. Galileo's relativity principle). The proof begins with explaining that the Lagrangian must only depend on the speed of the particle ($v^2={\bf v}^2$): $$L=L(v^2).$$ Hence the Lagrance's equations will be $$\frac{d}{dt}\left(\frac{\partial L}{\partial {\bf v}}\right)=0,$$ so $$\frac{\partial L}{\partial {\bf v}}=\text{constant}.$$ And this is where the authors say   Since $\partial L/\partial \bf v$ is a function of the velocity only, it follows that $${\bf v}=\text{constant}.$$  Why so? I can put $L=\|{\bf v}\|=\sqrt{v^2_x+v^2_y+v^2_z}$. Then $$\frac{\partial L}{\partial {\bf v}}=\frac{2}{\sqrt{v^2_x+v^2_y+v^2_z}}\begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix},$$ which will remain a constant vector $\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$ as the particle moves with an arbitrary non-constant positive $v_x$ and $v_y=v_z=0$. Where am I wrong here? If I am, how does one prove the quoted statement?  Proving that $L=\frac{m v^2}2$ ($\S$4. The Lagrangian for a free particle). The authors consider an inertial frame of reference $K$ moving with a velocity ${\bf\epsilon}$ relative to another frame of reference $K'$, so ${\bf v'=v+\epsilon}$. Here is what troubles me:   Since the equations of motion must have same form in every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, only by the total time derivative of a function of coordinates and time (see the end of $\S$2).  First of all, what does same form mean? I think the equations should be the same, but if I'm right, why wouldn't the authors write so? Second, it was shown in $\S$2 that adding a total derivative will not change the equations. There was nothing about total derivatives of time and coordinates being the only functions, adding which does not change the equations (or their form, whatever it means). Where am I wrong now? If I'm not, how does one prove the quoted statement and why haven't the authors done it? Does C1 code coverage analysis exist for Ruby? I'm currently using  to get C0 code coverage analysis for a rails project that I'm working on.  However, those results are practically meaningless- I have 100% coverage according to rcov (as it only covers C0 analysis) and I've barely written half the test cases for the functionality that exists thus far.  I'm used to the useful results from the code coverage in Visual Studio 2008 Team, which has C1 coverage.  Are there any tools that provide similar coverage for ruby? Have we ever observed two drosophila lineages that evolved reproductive isolation in labs? Background  The standard definition of species refers to the concept of . If two lineages are found to be reproductively isolated, then we consider these two lineages to belong to different species. My question concerns evolved reproductive isolation in Drosophila sp. following labs due to experimental evolution.  Question  Have we ever demonstrated that two Drosophila sp. lineages that could initially interbreed (in nature or in labs) evolved through artificial selection (and drift and mutations) in labs to finally not be able to interbreed anymore either due pre- or post- zygotic isolation (see )? Or, in other words, have we ever demonstrated that two drosophila lineages evolved to become different species (reproductive isolation definition) in labs experiments?  If not, have we ever observed some partial reproductive isolation such as inbreeding depression for example?    Note: This question is motivated by @LotusBiology that could not receive the answers she/he was waiting for because he/she failed to ask questions that are possibly answerable! So I wanted to ask this question that somehow addresses  question he/she asked (now on-hold)  For more information about the concept of species, please have a look at $P(x) = 0$ is a polynomial equation having at least one integer root, where $P(x)$ is a polynomial of degree five and having integer coefficients.  If $P(2) = 3$ and $P(10)= 11$, then prove that the equation $P(x) = 0$ has exactly one integer root.  I tried by assuming a fifth degree polynomial but got stuck after that.  The question was asked by my friend. java.sql.SQLException: Column count doesn't match value count at row 1 How to fix sculpting ridges? I ran into a problem I have never seen before and that is when I try to sculpt on a mesh, it sculpts like this:  Instead of this: I am using dyntopo with these settings:   I feel like theres a very obvious solution to this, but I have no idea what it is. I know I can only use the gym feature after hitting level 5, but are there any other privilege or rewards for the levels above 5?  For example, in Ingress, some items are only available for the higher level player. How about Pokemon Go? Framing a combination of cells in the notebook Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| &lt; 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the boundary (e.g. $\{ e^{it}, 0 \leq t \leq \pi/2 \}$ ), is it then true that $f$ is identically zero on the entire boundary?   I know that analytic functions that are zero at an accumulation point inside the domain of analyticity are identically zero throughout the entire domain, but I don't know what (if anything) can be said if something similar occurs on the boundary of the domain. Ralink MT7601U (148f:7601) Wi-Fi adapter installation Best way to extract a subvector from a vector? Suppose I have a std::vector (let's call it myVec) of size N.  What's the simplest way to construct a new vector consisting of a copy of elements X through Y, where 0 &lt;= X &lt;= Y &lt;= N-1?  For example, myVec [100000] through myVec [100999] in a vector of size 150000.  If this cannot be done efficiently with a vector, is there another STL datatype that I should use instead? How to get cross-references to multiple figures? Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.  My attempt: Let $T$ be the function $T: \mathbb{R}$ (over the field $\mathbb{Q}$) $\rightarrow \mathbb{R}$ (over the field $\mathbb{R}$). The set $\{1, \sqrt{2}\} \subseteq \mathbb{R}$ is linearly independent for the vector space $\mathbb{R}$ over $\mathbb{Q}$. Then, there must exist a linearly independent set $W \subseteq \mathbb{R}$ (over $\mathbb{Q}$) such that $\{1, \sqrt{2}\} \subseteq W \subseteq \text{span}(W) = \mathbb{R}$ (over $\mathbb{Q}$).  I have been told that the function defined as $T(1) = 1$ and $T(w) = 0$ for all $w \in \text{span}(W) \setminus \{1\}$ is additive but not linear, but I cannot see why this is? I can see why it is not linear, clearly $T(\sqrt{2} \cdot 1) = 0$ but $\sqrt{2} T(1) = \sqrt{2}$. But, why is $T$ additive? For example, $T(1+1) = T(2) =0$ but $T(1) + T(1) = 1+1 = 2$? Is there a mistake somewhere? Understanding the "Birthday Problem" Hide trivial comments I need to take a screenshot that records my Eclipse IDE menu item status. For example, if I clicked the "Project" menu and there comes a list of menu items. I need to record those menu items.  The problem is when the menu item list window (which works like a context menu window) has the focus, the screenshot function is not working. I tried everything on this  and nothing works as long as the menu items window is opened. Any mouse click will close the context menu window, so I cannot use mouse either. However if there is no context menu window opened then screenshots works fine.  How can I record the screen with context menu window displayed? My laptop is filled with viruses and Windows XP is just becoming impossible to work with.   I've been interested in Ubuntu for a while, so, would I be able to use something like Debian to clear my HDD and OS and then install Ubuntu and start fresh? How exactly does lighting affect combat? I understand that there are different types of lighting (dim, bright, complete darkness), but I haven't been able to find anywhere in the DMH or PHB that specifically describes how these different types of light sources affect combat.   What impact does complete darkness have on players being able to see the locations of the monsters in the room? What impact does complete darkness have on combat rolls against monsters? Could anyone point me to a book/page that contains some more detailed information on this?   Thanks! How to create a Customized Ubuntu Server ISO? I'd like to create a customized Ubuntu Server 12.04 ISO but couldn't find a clear approach on how to do it. I see that Ubuntu-builder is a very good app to create a customized Ubuntu Desktop, but what about Ubuntu Server? how to prove the chain rule? I have just learnt about the chain rule but my book doesn't mention a proof on it. I tried to write a proof myself but can't write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus.