I need to solve the following problem.   Show that if $f$, $g$ : $\mathbb R$ → $\mathbb R$ are continuous, periodic and $\lim_{x→\infty}$ $(f(x) − g(x)) = 0$, then $f = g$.  I have only used the definition of convergence so far. (Unfortunately, I couldn't go further for almost an hour.) For all $\epsilon$ there exists $ r\in \mathbb R$ such that $|{f(x)-g(x)}|\le \epsilon$ for all $x\geq r$  will you give me some hint?
If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in \Bbb R$. Now that $\displaystyle\lim_{x\to\infty}(f(x)-g(x))=0$.  Conjecture: $f(x) \equiv g(x)$.
Whenever I try to install anything with the Ubuntu Software Center, the following error appears.     Failed to download repository information      Check your Internet connection.      E:Unable to parse package file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_precise_multiverse_i18n_Index (1) The recent question  made me realize that I don't have any understanding of why one would upload to a preprint site.  (I don't work in a field where they seem to be common.)  Why would one be concerned about getting a manuscript on arXiv, if it's already under review with a journal?   Is it because of the long publishing timeline of journals?  (That could explain why computer science &mdash; which publishes much more in conferences &mdash; doesn't seem to do much with them.) Over on another network site, a helpful user corrected the grammar of .  The answer now says     I believe it's valid.   where I originally wrote     I believe it valid.   Is the original sentence gramamatically (or at least idiomatically) correct or do I absolutely need a form of 'to be' here? (What's more, is 'to be' a copula or an auxiliary verb if used like this?)  I am quite certain that I have seen sentences like the original one in the past, but I am uncertain where. Given that I am frequently exposed to the somewhat made-up English of authors in the fields of sci-fi and fantasy, I might have fallen into a trap here.  Edit: I don't doubt that the correction is valid and enhances readability. I have never had this problem before. But after seeing some questions posted on ELU, I started to doubt. I have been through links1, pages of grammar books2, a lot of questions3 posted here and ELU. I now still can't seem to understand how the past perfect is exactly used. The references sometimes even contradict each other.  What I had understood so far was      The past perfect is not necessary if we are not 'going back' to the earlier past, and simply moving from one event to another.   but I then saw a lot of other 'rules' including      If the sequence is clear, the past perfect is not needed.   which is slightly different, less 'narrow' and can be applied in more situations than the first ruleA, and      The past perfect is optional only when talking about an action at a specific time.   This is confusing. Can anyone here clarify things out please?    A For example, for  Doing the survey [completed] -&gt; Showing the result      We showed the result of the survey we did.      If the second rule is applied, the sentence above is fine (the sequence is clear), but not if the first rule since we are not moving from any event to the next and possibly even going back.    1  and   2 Practical English Usage (2005) by Michael Swan, Collins Cobuild Student's Grammar (1991). Upper-Intermediate Cutting Edge.  3 including , , ,  etc. Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA?  I have looked at many sources online and have not found anything that had the right combination of rigor and accessibility.    I have looked into The Elements of Statistical Learning, which was suggested in a comment on a question asked on Cross Validated, , but I don't think that this reference does the topic justice (it's too brief to do so, and doesn't provide much theory on the subject).  From what I've read, PLS exploits linear combinations of the predictor variables, $z_i=X \varphi_i$ that maximize the covariance $ y^Tz_i $ subject to the constraints $\|\varphi_i\|=1$ and $z_i^Tz_j=0$ if $i \neq j$, where the $\varphi_i$ are chosen iteratively, in the order in which they maximize the covariance.  But even after all I've read, I'm still uncertain whether that is true, and if so, how the method is executed. Example 1) Let's say I have some random creature with protection from white, and my opponent plays , a white card with the effect "Destroy all creatures." Can my creature with protection from white be destroyed by a white card that isn't actually targeting it specifically?  Example 2) I was playing a match with a friend where I had  on the field against his 3 zombies. Is he able to declare his zombies as blockers even though Elite Inquisitor has "protection" from zombies?  If he can declare them as blockers, does protection mean that no damage is dealt by the zombies to  Elite Inquisitor? If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.  It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.  Regards We have to shift our web-site to a new domain due to some unavoidable circumstances. The portal has been up and running there since 2009.  It has good index in various search engines and we are getting a good number of page hits on daily basis.  Losing the search hits and index are our big concern. We are not sure how best we can handle this.  What all can be done to preserve as much as possible while shifting to new domain name. @class vs. #import Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.  $V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$  $W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$  $U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$    I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't? How do I add a new language? The aftermarket ROM I use seems to have the language list parred down to just a couple selections, probably to save space.  I would like to add a language back into the system. I can already set the locale using programs such as  so that applications with my language support start with the right locale code, however that doesn't add missing system languages. Having the system in another language would be nice, but the main thing is getting the dictionary files for the keyboard auto-complete. How does one go about adding the language support files for an input language to a system or ROM?  I can access other ROM's or the AOSP project to rip things out, but I don't know where to start. What files do I need and what modifications need to be made? Is there an easy way to do this with say a language-pack or is it hackery all the way?  My ROM is fully rooted and I have adb access if necessary. It's Android 2.3.7 () and I'm using the stock Android keyboard. The language I want is Turkish, although unless somebody has a download to point me to it's mostly the process I'm looking for here. Ubuntu boots on terminal only, how do I recover the GUI desktop? are more efficient than ordinary upright bicycles, and  are even more efficient. Then why do we see them so rarely on the road? Is it because they are expensive, or are there some non-financial, significant disadvantages? What's wrong with this? overbrace/underbrace but with an arrow instead Is there a similar command to \overbrace{foo}^{bar} but instead of braces, an arrow pointing up/down?  EDIT: Here's a little demonstration of how I'd expect something like this to work:    And possible syntax would be \overarrow{a=b}^{\text{theorem X}} Integrating MySQL with Python in Windows How to export / dump a MySql table into a text file including the field names (aka headers or column names) A good title helps your question get the attention it deserves.  What goes into a good title? An invalid argument, the conclusion of which is a tautology Why it is not possible to have such argument?  I found some textbook Introduction to Formal Logic and this is from one of the first excercises. This is marked as impossible, but I do not understand why. I watched a video on  and read a few  which say that human beings see 3D because we have two eyes. But does that mean when I close one of my eyes, I should see 2D? It doesn't happen so. Why am i still able to be understand depth? Would a person with one eye not be able to perceive depths?