Prove that $1 + nx ≤ (1 + x)^n$ , for all real numbers $x &gt; −1$ and integers $n ≥ 2$ by mathematical induction.  My attempt (1) $n=2$ $x &gt; -1 \Rightarrow (1+2x &gt; -1) \land (x^2 \ge 0)   \Leftrightarrow (1+2x) \le (1+x)^2 $ Hence p(2) is true.  (2) We must show that for all integers $k \ge 2, p(k) \Rightarrow p(k+1)$   Suppose for inductive hypothesis, $k \ge 2, p(k)$. That is $1 + kx ≤ (1 + x)^k$ Then, $1 + kx + x = 1 + (k+1)x ≤ (1 + x)^k + x$    I should get $1 + (k+1)x ≤ (1 + x)^{k +1} $. It seems when I multiply each side by (1+x), the left side gets complex and far away from desired inequlity. On the other hand, when I add 1, it doesn't help since we have to get $(1+x)^{k+1}$ on the right side, $(1 + x)^k + x+1$ doesn't increase the exponent of $(1+x)$.  What should I do next to get out of $1 + kx + x = 1 + (k+1)x ≤ (1 + x)^k + x$?   $1 + kx + x = 1 + (k+1)x ≤ (1 + x)^k$
I'm asked to used induction to prove Bernoulli's Inequality: If $1+x&gt;0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far:  Let $n=1$. Then $1+x\geq 1+x$. This is true. Now assume that the proposed inequality holds for some arbitrary $k$, namely that $$1+x&gt;0\implies (1+x)^k\geq 1+kx,~\forall~k\in\mathbb{N}\setminus\{1\}$$ is true. We want to show that the proposed inequality holds for $k+1$. Thus multiplication of $(1+x)$ on each side of the above inequality produces the following result: $$(1+x)(1+x)^k\geq (1+kx)(1+x)\implies (1+x)^{k+1}\geq 1+x+kx+kx^2\cdots\cdots\cdots$$  I'm not sure where to go from here.
How can I manage the "Open with Explorer" button based on user permissions? How do I determine if an array is initialized in VB6? Passing an undimensioned array to the VB6's Ubound function will cause an error, so I want to check if it has been dimensioned yet before attempting to check its upper bound.  How do I do this? In episode 10 of Full Metal Alchemist: Brotherhood, during the events in the Research Lab 5, Edward Elric is severely hurt. Later in the hospital, he writes down everything he had come to know, and he discusses with Alphonse, Hughes, and Armstrong about what to do with the information to continue the investigation.  At this point, King Bradley arrives with lemon into the room. He finds out what they are doing but acts very calm and chilled out and allows them to continue investigation while being careful.  Why did he not get Hughes and Armstrong killed at that point? Or at least tell them to not investigate at all? Why does volatile exist? What does the volatile keyword do? In C++ what problem does it solve?  In my case, I have never knowingly needed it. Limit point compactness implies compactness. I have a question about the following counterexample, why  $U_n = \{n\} \times Y$ has not finite collection covering $X$?   Is it because if you take a finite set $S\subset U_n$ then it's impossible to have $X\subset \bigcup S$ since X is infinite?  Counterexample:  Let $Y$ consist of two points; give $Y$ the topology consisting of $Y$ and the empty set. Then the space $X = \mathbb{Z}_{+} \times Y$ is limit point compact, for every nonempty subset of $X$ has a limit point. It is not compact, for the covering of $X$ by the open set $U_n = \{n\} \times Y$ has not finite collection covering $X.$ I have installed a vlc in centos 6.5. I want to run it as root. But I get the following error,      "VLC is not supposed to be run as root. Sorry. If you need to use   real-time priorities and/or privileged TCP ports you can use   vlc-wrapper (make sure it is Set-UID root and cannot be run by   non-trusted users first)." How can I make a peer-to-peer multiplayer game? How can I make a p2p multiplayer game? I would like to have a server-less multiplayer game. But then, how all the clients know each other?  Why the p2p-protocol is so famous in file transfer but not in multiplayer games? I am a new user for Ubuntu. I downloaded 13.10 yesterday and made a bootable USB with universal USB installer on my Lenovo IdeaPad Z580 laptop which is now using UEFI; everything appeared OK.  When I booted from the LiveUSB I got the choices of trying or installing Ubuntu but both of them keep giving me these error:  [ 1.929082] kernel panic-not syncing vfs unable to mount root fs on unknown-block (0.0)   Please tell me what is going wrong? Prove that if every node in a simple graph $G$ has degree $3$ or higher, then $G$ contains a cycle with a chord. Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ Sphere rolling up a ramp In this question, I did energy conservation, with the states being the initial one and the one where it is at the top , and I got $$ .5 I w^2 + .5 mv^2 = mgh$$ However, in  the answer key they have done, $$ .5 mv^2 + .5 Iw^2 = .5 Iw^2 + mgh$$ Now why would the k.e be turned to p.e but not the rolling energy?? Orthogonality and linear independence [Theorem] Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent.     Also, orthogonal set and linearly independent set both generate the same subspace.  (Is that right?)    Then orthogonal $\rightarrow$ linearly independent but orthogonal $\nleftarrow$ linearly independent is that right?    One more question. For T/F, Every orthogonal set is linearly independent (F) Every orthonormal set is linearly independent (T) Why? How can I see flags I cast from all sites? Diffeomorphism group of the unit circle I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and $\operatorname{Diff}^-(\mathbb{S}^1)$, the diffeomorphisms that preserve or reverse the canonical (counterclockwise) orientation, respectively.  Question 1: How does one prove that?  Question 2: Given $\Phi, \Psi \in \operatorname{Diff}^+(\mathbb{S^1})$, can one construct an explicit path joining them?  I'd be satisfied already with a proof that we can join $\Psi, \Phi \in \operatorname{Diff}^+ (\mathbb{S}^1)$, without giving the path explicitly. I tried the obvious path, $t \mapsto \dfrac{t\Phi + (1-t)\Psi}{|t\Phi + (1-t)\Psi|}$, but that doesn't seem to work.  Thanks. How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? I am looking at collecting spatial information from a range of stakeholders to identify opportunities within a landscape.  However, the data that we are collecting are not presently mapped and so we wish to use an interactive method to get this information onto a map.  Which online websites/methods would people recommend for circulating to a non-technical audience to try and get them to map what they know?    Where the emphasis is on ease of use by the stakeholder. What is a current replacement of gnome-open command (generic open of files based on the type)?  Before:   gnome-open mydoc.pdf  # opened PDF in default application   Now:  gnome-open The program 'gnome-open' is currently not installed.  You can install it by typing: sudo apt-get install libgnome2-0 Need to understand why my Schengen visa was refused Looking for title/author of SF novel about malevolent creatures/gods residing inside stars IVP Differential Equation I have stumbled across a very old exam question from my linear algebra course and the solutions are not available. I was wondering if my working/logic is correct and if any improvements can be made.     Let  $\ B=\begin{pmatrix}     1 &amp; -1 &amp; 4 \\     6 &amp; -7 &amp; 2 \\     -3 &amp; 1 &amp; -6       \end{pmatrix}\ $ and $\ \vec{v_1}=\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$.      Part (i): Calculate $(B+5I)\vec{v_1}$ and $(B+5I)^2\vec{v_1}$.   Using simply matrix multiplication, I calculated $(B+5I)\vec{v_1}=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $(B+5I)^2\vec{v_1}=\vec{0}$.     Part (ii): Find all eigenvalues and eigenvectors of B.   From our calculations in part (i), we can see that $\ \lambda=-5,-5$ are two of the eigenvalues, with corresponding eigenvector $\ \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and generalised eigenvector $\ \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$. From the trace of B, the remaining eigenvalue is $\lambda=-2$ with corresponding eigenvector $\begin{pmatrix} -2 \\ -2 \\ 1 \end{pmatrix}$.     Part (iii): Giving reasons, write down a basis for the generalised eigenspace $GE_{-5}$ of B.   I am unsure of this part of the question. If I had to guess, I would say a basis for $GE_{-5}=\left\{\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix},\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}\right\}$. Is this correct?     Part (iv): Solve the initial value problem $\vec{y'}=B\vec{y}$ where $\ B=\begin{pmatrix}     1 &amp; -1 &amp; 4 \\     6 &amp; -7 &amp; 2 \\     -3 &amp; 1 &amp; -6       \end{pmatrix}\ $ and $\vec{y}(0)=\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$.   Well, the solution to this IVP will take the form $\vec{y}=e^{tB}\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$. I noticed that $\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$ was a generalised eigenvector and hence we can write $\vec{y}=e^{-5t}\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}+e^{-5t}\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}t$. Thus $\vec{y}=e^{-5t}\begin{pmatrix} 1+t \\ 1+2t \\ -1-t \end{pmatrix}$