Anyone can give me a hint please?  "Show $\mathbb{Z}^{\mathbb{N}}$ is not a free-abelian group"
Why isn't an infinite direct product of copies of $\Bbb Z$ a free module?   Actually I was asked to show that it's not projective, but as $\Bbb{Z}$ is a PID, so it suffices to show it's not free.   I am stuck here. I saw some questions in SE, but there is no satisfactory answer at all.
Is there a fast way to get to the bottom of the Death Star? In Tiny Death Star, you can get to the top of your tower by tapping on the top of the screen while avoiding the "Add Level" button. However, the only way I can find to go to the bottom of the tower is to wait for someone to want an elevator ride, and tap on the button to give them that ride. Is there any other hot-spot for getting to the bottom of your Death Star? If $A$ is compact, is then $f(A)$ compact? On the mobile web after new privilege is earned, reputation block contains some design issues:   no margins poor text color inside bar         By tapping on "track the next one" link, page have been refreshed:   no margins empty block in the middle obscure content         Only after manual refreshing, page looks fine again:        It is hard to reproduce because you need to earn next privilege which takes a lot of time. But I believe it's no too hard for developers :) For most OpenID options, it doesn't much matter if you pick the "Sign up" versus the "Log in" tab on the sign in page.  Whether you have an account on the site or not, it just does the right thing. Not so if you are using the Stack Exchange OpenID service:    (Bonus bug: the reCAPTCHA box is a bit too wide for the expanded "Sign up using Stack Exchange" container.)  Since we already have collected an email and password (well, two copies of the password), could we not check to see if they match and automagically log the user into their SE OpenID?  If not, the error needs to be reworded somehow to explain that while the email might have been used to create a Stack Exchange OpenID, that doesn't mean they can't log in with that OpenID to create an account.  Counterintuitively, they need to use the  even if they've never had an account there before.  The  link adds to the confusion since it sends an email that leads off with:     Someone requested an account recovery on The Workplace Stack Exchange for jericson@stackexchange.com, but we don't have an account on this site that matches this email address.   (For people with accounts on other sites and/or Stack Exchange OpenIDs, this paragraph is followed by other instructions that would probably work if followed. By that time I imagine many users are at wit's end.)  To simplify: If a user already has a Stack Exchange OpenID (likely because they have an account on another site) they should be able to use the  to sign up for an account or there should be clear instructions on what to do next if they fail. How can I make an object show up on different layers How do you secure wood framing to a concrete basement floor? My case here is that I was writing something for school, and my teacher explained an error which I really can't see. She referred to articles about &quot;dangling participles&quot;, which was previously unknown to me. I find it hard to relate the examples I've found about it to my situation. So this is a text I've written and the bold part is where the problem arises: (You could read only the bold part if you please)  Wandering, and lost in his thoughts, there is a deep despair occupying the mind of this man. He is barely aware of his surroundings, except for those few moments when the loud blissful voices break through the protective barriers, developed by his immense sadness and grave frustration, into his mind. ”How can they all possibly be so happy?” he thinks to himself. Staring blindly, with his head low, as he walks in a slow pace, whispering ”Mary, Mary, my Mary,” over and over again. Yesterday was her funeral. She had passed away from a severe type of cancer, at a relatively young age. It was his wife. They had been together since high school and were greatly attached to each other. Having spent nearly half his life by her side, we cannot but imagine his grief. Indeed, he could hardly picture a life without her. So, alone he roams in the midst of all the seemingly incessant cheering and noises, drowned by his sorrow.  Her comments on that part was:  we is slightly out of place here – see my comment below These few sentences are about him and his emotions, so ‘we’ seems out of place. If the sentence starts: ‘Having spent nearly half his life by her side’, the reader expects it to continue: ‘he’. After all we have not spent half our lives by her side.  Is she right? If so, could anyone try to explain why? I did apt-get upgrade it never finishes at this point:  Searching for GRUB installation directory ... found: /boot/grub Searching for default file ... found: /boot/grub/default Testing for an existing GRUB menu.lst file ... found: /boot/grub/menu.lst Searching for splash image ... none found, skipping ... Found kernel: /boot/vmlinuz-3.2.0-24-virtual Updating /boot/grub/menu.lst ... done  You already have a NoLOADER configuration in /etc/grub.conf Running boot loader as requested Running /usr/sbin/grub  ...    The system does not freeze (Ctrl-Z works, I get my console back) just that that /usr/sbin/grub seems to never complete.  What should I do? Some time ago I closed my front door (rather forcefully) while the deadbolt was sticking out partially.  This loosened the deadbolt, and the top screw no longer held. In addition, it cracked the door on the interior side. I've attached a picture showing the damage.   The door works mainly, though it can be difficult to lock sometimes, especially when the weather gets colder.  I'm currently renting, and will soon be moving out of this house, once I buy a house. I would like to repair this door myself, and I'm hoping I don't have to replace the entire door.  Is this something that can be repaired? If so, how? I realize that a new deadbolt is probably needed, but how can I repair the wood damage around it? RegEx match open tags except XHTML self-contained tags How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$ Let $f\colon(0,\infty)\to\mathbb R$ be differentiable on $(0,\infty)$ and suppose that $\lim\limits_{x\to\infty} f'(x)=L$.         a. Show that for any $h&gt;0$, $\lim\limits_{x\to\infty} \frac{f(x+h)-f(x)}h=L$.      b. Show that $\lim\limits_{x\to\infty} \frac{f(x)} {x} =L$.   this is a differentiation question.  i don't know how to solve this question  Because it is Korean, it is difficult to deal with this cite. So for me, it is not easy to write.  thank you Number of labeled trees which have one vertex of degree 4 How many labeled trees which have one vertex of degree 4 can we have? A tree with $10$ vertices has $9$ edges, so the sum of the degrees of the vertices is $2\cdot 9=18$. If there is one vertex of degree $4$, the sum of the degrees of the other $9$ vertices must be $14$. I don't know what to do next. Should I count all possible vertex degrees? What is a bus error? What does the "bus error" message mean, and how does it differ from a segfault? Meta's tag pages inaccessible due to self-redirect loop Tag pages (route /questions/tagged/tagname) redirect to themselves indefinitely:  $ curl -D - http://meta.stackexchange.com/questions/tagged/featured HTTP/1.1 302 Found Cache-Control: private Content-Type: text/html; charset=utf-8 Location: /questions/tagged/featured X-Frame-Options: SAMEORIGIN Date: Fri, 05 Jul 2013 16:47:42 GMT Content-Length: 143  &lt;html&gt;&lt;head&gt;&lt;title&gt;Object moved&lt;/title&gt;&lt;/head&gt;&lt;body&gt; &lt;h2&gt;Object moved to &lt;a href="/questions/tagged/featured"&gt;here&lt;/a&gt;.&lt;/h2&gt; &lt;/body&gt;&lt;/html&gt;   This only happens on Meta, so far, but only applies to any meta-tag that doesn't have a dash in it; those with a dash seem to be exempt, as do invalid tags:    redirects.  does not.  does not. "Please do not repeat the mistake" and "Please do not repeat the mistake again" Please do not repeat the mistake. Please do not repeat the mistake again.  Should the again be omitted? “Repeat” means “do again”. So to “repeat again” means “to do again again”. Is the second sentence grammatically wrong? Is there an exception to the prohibition against ending a sentence with “ ’s ” at work here? The ’s can be used as a contraction representing a weak, unstressed word that is not pronounced.  It allegedly cannot occur in sentence final position.        She is not ready, but he is.   She’s not ready, but he is.   She’s not ready, but *he’s.      The last one is not grammatical there.  Similarly, here the last one is not grammatical:        He has not started yet, but she has.   He’s not started yet, but she has.   He’s not started yet, but *she’s.       claims this is because you cannot end a sentence with a weak form.  It backs up its point with reasonably scholarly documentation.  However, this does not seem to be invariably true.        Shall we go to the movies?  Yes, let us go.   Shall we go to the movies?  Yes, let’s go.   Shall we go to the movies?  Yes, let’s.      Unlike the third example in the first two sets, here in this case the third example is indeed grammatical, despite ending with a weak form.  Is this an exception to the rule, or is there another rule at work here? Clearing all cookies with JavaScript Named blocks to limit variable scope: good idea? Short connection at IAD (Dulles international airport) I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? Dashed line and Wireframe