Given $\lim a_n =a$ and $\lim b_n =b$,  show $\lim\frac{1}{n}\sum_{k=1}^{n}a_k b_{n+1-k}=ab$  As you can see, two limit are expanding in two different direction. I have no idea about how to do it.
Let $\{x_n\}$ and $\{y_n\}$ be sequences of real numbers which converge to $\ell$ and $m$ respectively. Show that  $$\lim_{n\to \infty} \frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}=\ell m$$  This is a problem I found in a book, it seemed to be routine, but now I can't solve it.  What I tried :     Let us take some $\epsilon&gt;0$. Now after a stage $N$, both $|x_n-\ell|&lt;\epsilon$ and $|y_n-m|&lt;\epsilon$. Now I wanted to break the sum $$\left|\frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}-\ell m\right|$$in two parts, one upto $N$, and other one for $N+1$ to $n$. Now we need this to be arbitrarily small. But here I am lost. I don't know how to proceed anymore. Because when I take $x_i's$ to make up a finite sum, then the $y_j's$ are there too with $j$ very large. I think this is the main idea, but unable to do anything further. Can someone help me? Thanks.
Using trickle to limit the bandwidth of a VirtualBox VM I'm trying to limit the bandwidth of a virtual machine running via VirtualBox.   trickle -d 5 virtualbox doesn't work, I believe because of an issue with child processes (trickle -d 5 sudo apt-get upgrade doesn't work either)  How can I limit the download/upload bandwidth of VirtualBox's VMs? Convergence of the arithmetic mean Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with limit $a \in \mathbb{R}$. Show that the arithmetic mean given by: $$s_n:= \frac{1}{n}\sum_{i=1}^n a_i \tag{A.M.} $$ also converges to $a$.  I have read:  but unfortunately the answers there don't help me much because I don't understand their substitutions and most of it all, why their substitutions seem to work.   What I know from the problem is that since $a_n$ is convergent and $\epsilon &gt;0$ is given, I can say that:   $$\exists N_1 \in \mathbb{N}, \forall n \geq N_1: |a_n-a|&lt;\epsilon_1 $$ I also know that since $a_n$ is convergent, it is bound, so $(a_n) &lt; M, \ \forall n \in \mathbb{N}$   I need to show that: $$\exists N_2 \in \mathbb{N}, \forall n \geq N_2: |s_n-a|&lt; \epsilon_2$$ I started as follows: $$ \left|s_n-a \right| = \left|\frac{1}{n}\sum_{i=1}^na_i -a\right|= \left|\frac{1}{n}\sum_{i=1}^m(a_i-a)+\frac{1}{n}\sum_{i=m+1}^n(a_i-a) \right| \\ \leq \frac{1}{n} \sum_{i=1}^m|(a_i-a)|+\frac{1}{n}\sum_{i=m+1}^n|a_i-a|$$ I believe to understand that the left sum after the $\leq$ is finite, bound and doesn't depend on the values that $n$ takes on. However, I don't understand where all the substitutions come from and make this proof so seemingly easy to complete.   Is there a general idea I can follow to complete such proofs? Because I know that the last step is to show that the given sum is smaller than $\epsilon$. I also know that I should bring the condition $a_n &lt; M$ into place somewhere, but I don't know where.  If I choose an $n \geq N: |a_n-a|&lt; \epsilon'$ what does that tell me about $|a_i-a|$? I know that they are seemingly the same just with a different index. "...programs that each perform..." vs "...programs that each performs...": which is correct? Replaced wrong car in garage in GTA Online I was asked to replace a car in order to put another car in my garage. However I thought I would be able to swap the car I wanted in and take out the other car that was already in there so I could bring it to my other garage. Can I get my car back? Parenthetical pluralization of words ending in '-y' Real Analysis, Existence of a Limit Boundedly:$|\int_0^b\frac{sin(ax)}{x}dx | \le c$ I need to show that there exists a $c$ such that $|\int_0^b\frac{sin(ax)}{x}dx | \le c$, where $a$ is any real number, and $b$ is any real number greater than 0. Also I need to show that the smallest $c$ possible is $c=\int_0^\pi\frac{sinx}{x}dx$. I know that $\int_0^\infty \frac{sin(ax)}{x}dx = sig(a)\frac{\pi}{2}$ where $sig(a)$ literally just means positive one or negative one, depending on the sign of $a$. I think the fourier tranform of the characteristic function on the interval $(0,a)$ is $\frac{sin(ax)}{x}$. After that, I am unsure about how to proceed. Thanks for your help! I have been trying to achieve a true mesh displacement on meshes for quite some time, without ever getting it to work.  If I understand it correctly, this experimental feature would allow me to easily displace the vertices of a mesh - or to create new mesh vertexes where needed - to get a displaced mesh in the result, based off an image, but without using a bump map (simpler, faster?).   This all sounds very exciting in theory, but I am having a really hard time finding examples or tutorials that are both recent and whose links function. So I think there is the need for a question here which will call upon detailed and recent answers to bridge that gap.  I'll start with a simple ico sphere:    Then, , I will change the Feature Set to the Experimental one:    I then add a pink granite image to the mesh:    Mark Seams, Unwrap, add image in the UV Editor:      and finally, in the Object Data panel, change the Displacement method from Bump to True as explained :    no result yet:    if I click "Use Subdivision" this is what I get (so I won't click it right now):    Looking into the Node Editor, nothing seems to have changed with the Displacement of the mesh:    so I tried to replicate the node setup seen in  of  (Linear isn't an option, Clip, Single and Repeat are):    ... but the result is a white ICO Sphere, similar to Image 1.  So, can the "Experimental Feature Set" be used to create actual displacement, and modify a mesh?  This is what I'm trying to obtain - but with the image of the texture actually displaying on the mesh (image from ):    .blend file: What is the probability one card from each suit will be represented when 5 cards are dealt? Why *not* parse `ls` (and what to do instead)? What is object serialization? Order of "not" with infinitive This is one thing that keeps bugging me, and maybe there's a direct answer. Grammatically, which one is more correct of these two?  Does it make a difference?  I tried not to do that. I tried to not do that. The question I want to ask is related to the answer of this question:  .   My aim is to separate my bibliography into two parts : one part for a personnal bibliography, and a second one for the regular bibliography.  To do that, I use the technique of the post I cited :  \documentclass{article} \usepackage[T1]{fontenc} \usepackage[french]{babel} \usepackage[style=alphabetic,maxnames=6,natbib=true]{biblatex} \bibliography{biblatex-examples}  % Variants of each could be added \newcommand{\firstname}{Donald~E.} \newcommand{\lastname}{Knuth}  \DeclareBibliographyCategory{byname}  \DeclareIndexNameFormat{byname}{% Test could be refined   \ifboolexpr{ test {\ifdefstring{\lastname}{#1}}            and test {\ifdefstring{\firstname}{#3}}               } {\addtocategory{byname}{\thefield{entrykey}}} {}}  \AtDataInput{%   \indexnames[byname]{author}}  \begin{document}  \section{Knuth's books}  \begin{refsection}  \nocite{*}  \printbibliography[category=byname,heading=none]  \end{refsection}  \section{Not Knuth's books}  \printbibliography[notcategory=byname,heading=none] \end{document}   The problem is that only the second bibliography appears, and when I remove the refsection, then all the bibtex entries of my input files are printed, separated as they should.  EDIT: the aim is to have all publications (even the ones not cited) by author knuth in the first section and all the other cited publications in the second one. How can I keep low resolution textures pixelated? I have a problem. I am going to make a minecraft animation in blender. But when I put a minecraft texture on a 3d cube that is 16x16x16 it is very low resolution is there a way to get the texture in a better resolution or is there a way to make blender make the texture better?  I hope you have an answer. Thanks in advance How are unmarketable market orders (other side of the order book is empty) matched with incoming orders? Rest mass of the photon Why is the rest mass of the photon is taken to be zero?  If rest mass of the photon is non zero, then it leads to infinite energy of the photon (by $E=mc^2$ equation) and there is a violation of conservation of energy. Is this a reason?  Also, is there any thing called the relativistic mass of the photon? My homework has this problem:     Prove that there exist irrational numbers $x$ and $y$ such that $x +  y$ is rational.   There is an easy solution that I found on mathbitsnotebook.com:     \begin{align} \left(2+6\sqrt{7}\right) + \left(-6\sqrt{7}\right) = 2.\end{align}   Can we prove the same without using subtraction? Primefaces growl is not working in all the pages Remove HTML Tags from an NSString on the iPhone How to remove desktop environments? I installed few environments that I wanted to try out on Ubuntu 12.04, but none of them worked at all. It could be that I installed them all at the same time, meaning the OS didn't get a chance to work everything out, but either way, they didn't work. I would now like to remove them.   The one's that I installed are Xubuntu/Xfce, Lubuntu/LXDE, Kubuntu/KDE, Fluxbox, OpenBox, cinnamon, and MATE. I installed them. Everything seemed to have been working properly until I actually tried to use the shells and nothing loaded at all. Except for Fluxbox, I think that one worked. I want to know of any way to repair or perhaps just remove the packages entirely. I might have already removed them because I did the apt-get remove command on all of them, but they were still in the list on the login screen. Why GCP Kubernetes Node host location for europe-north1 seems to be in US?