For what values of $\alpha$ would the given conditions hold? I don't know how to proceed.
I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks
This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks. Word order: "10 minutes more" or "10 more minutes"? Create a shortcut for URL? Ring isomorphism (polynomials in one variable) Let $X$ be adapted integrand, for which we know that $E[\int_0^T X_s^2 \, ds]&lt; \infty $. Let $0≤s&lt;t≤T$ and we have some $W∈F_s$ ($F_s$ is $\sigma$-algebra) and we know $E[\int_s^t W^2X_u^2 \, du]&lt;\infty$. I have to show  that $\int_s^t W X_u \, dB_u=W \int_s^t X_u dBu$.  I know that for simple integrands this follows from definition. We also know that for $X$ there exists sequence of elementary integrands $X^n$ such that $E[ \int_0^t (X^n_u-X_u)^2 \, du]$ converges to $0$, when $n \to \infty$. From this one can see that it also holds $E[ (\int_0^t (X^n_u-X_u) \, dB_u)^2]$. I wonder if I can show that it  holds $E[ |(\int_s^t W (X^n_u-X_u) \, dB_u)|] \to 0$ as $n \to \infty$ . (I think I should use Cauchy- Schwarz inequality: $E(XY)^2 \leq E(X^2)E(Y^2)$, but not sure where. If I could show this convergence in $L^1$, it would be enough for my proof. If I rewrite it as $E[E[|(\int_s^t W(X^n_u-X_u) \, dB_u)|  \mid F_s]]$ I don't know if I can put $W$ out of integral. Would it be correct? Any help would be really very appreciated. Thanks. This song by The Rolling Stones, "(I Can't Get No) Satisfaction". As a non-native speaker, I always wondered what's the exact meaning of this phrase?   Is it "I'm not getting any satisfaction" (this seems to be the most widely met translation; double negation used where single negation should have been used). Or is it "I want my satisfaction, I cannot leave without getting some" (makes much more sense to me).   Or is it something else? I've always had troubles understanding double negation as applied to English. First I was taught it's a mistake to use double negation at all, and then I see it everywhere. Let $V,W$ be two vector spaces over a field $F$ and let $f:V \to W$ be a linear mapping. I have going over some facts about vector spaces (not necessarily finite dimensional) and I have been trying to apply the rank nullity theorem to derive a simple lemma.  I have not been able to complete the proof and was wondering if I need to use something other then $\ker f + \mathrm{Im}\; f = \dim V$.  How do we show that for ever vector subspace $K \subset W$, $ \dim(f^{-1}(K)) = \dim(K \cap \mathrm{Im}\; f) + \dim( \mathrm{Ker}\; f)$? Dumbed-down explanation how scientists know the number of atoms in the universe? It is often quoted that the number of atoms in the universe is 10$^{70}$ or 10$^{80}$.  How do scientists determine this number?   And how accurate is it (how strong is the supporting evidences for it)?  Is it more likely (logically >50% chance) that the numbers are right, or is it more likely that the numbers are wrong? $n$th derivative of $e^x \sin x$ Can I exclude spam domains from my search results? Sometimes when I search for something, there are plenty of links to sites with no content (those who only have the premade templates which the search terms), spammers, and leechers (as in copying content from other sites, like B r o t h e r s o f t who leeches on SoftPedia.com).  How come they have such a high ranking without any original content (furthermore, why are they not simply banned from the results)?  And also, is there some way of filtering them out? Like for example completely exclude a domain from search results? What is "function*" in JavaScript? Windows 7 booting fail with error code 0xc0000225 I have an HP laptop Pavilion DV4. Last night, I installed Ubuntu and there was an problem about partitioning the hard disk, and I think I deleted the Windows boot record.  I didn't burn a recovery disc but there is built-in recovery partition on the hard disk.  All I want to do is to perform a Factory Restore but I need recovery disc. I could burn that disc from the recovery partition using HP recovery manager, but can't burn now. I can't also boot HP recovery manager using F11 key, because it has the same boot error.  I want to boot Windows 7 and want to burn recovery disc. How can I boot into my old windows 7. If I can boot it, I can burn Recovery Disc and I will make Factory Restore if it's needed.  I didn't make any changes to windows 7 partition but I did a problem about Boot Record.  There is a system partition on hard disk with the size of about 200 MB.There is BOOT folder too.  Installation of Ubuntu also failed, so I'm booting from Live Ubuntu USB. I've got a rather large number of files on my Windows drive that have a colon in their name. (These came from unpacking some Unix archives.)   When I try to delete them, Windows (XP) complains that the file does not exist, and refuses to delete it. This happens when trying to delete or rename it from explorer or the command line. Still, chkdsk does not complain about those files nor fix the problem.   Any ideas on how to get rid of them? How can I determine if a C++ object has been deallocated? I am trying to use netcat to send a simple message over TCP, e.g. I want to send the digit 1.  So I understood that I write the following in the terminal, after installing netcat  netcat [ip-address] [port]  But how do I specify the message to be sent? i.e. where to I write "1" ? Trying to find some simple SQL Server PIVOT examples.  Most of the examples that I have found involve counting or summing up numbers.  I just want to pivot some string data.  For example, I have a query returning the following.  Action1 VIEW   Action1 EDIT   Action2 VIEW   Action3 VIEW   Action3 EDIT     I would like to use PIVOT (if even possible) to make the results like so:  Action1 VIEW EDIT   Action2 VIEW NULL   Action3 VIEW EDIT     Is this even possible with the PIVOT functionality? Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. Update Manager doesn't ask for a password Oneiric update manager no longer asks for a password to update already installed software.  This is a change in behavior from previous versions.  Is there a way to revert this behavior and get the Update Manager in Oneiric to ask for a password again?  Thanks! How to properly unwrap my mesh? Is there a way to automatically unwrap my character? If not, is there a tutorial or something? Does there exist a set L of pairwise disjoint lines, each line in an axis direction, in $\mathbb{R}^3$ such that their union is equal equal to $\{ x : x \text{ has at least one irrational coordinate}\}$?