I am solving tutorials of an institution which I am not a student as our faculty doesn't gives any.  Please give me some hint in solution of this problem.  Problem is - let X be a normed linear space.  Prove that X is a Banach space iff $\,\{ x: \|x\| = 1 \}\,$ is complete.
Let $X$ be a normed vector space.   Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if and only if $X$ is Banach.     I have found the first part in a book, but I do not how to prove part 2. Thanx regards
Equal simple field extensions? I have a question about simple field extensions.  For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$? What are the odd of a single coin toss after many consecutive ones? I have little knowledge of probability and have recently been thinking of a math problem I am sure you guys could answer  If for example I tossed a coin ten times and got all heads, would the odds of the next coin toss beeing heads be 50 50 or s very low as the odds of getting 11 in a row is tiny?   Thanks In The Force Awakens, Finn and Poe steal and escape in a TIE wearing no personal life support apparatus. TIEs have been upgraded with shields, hyperdrives (for "/sf"), weapons etc, but as far as I know they weren't upgraded with built in life support.   Finn and Poe should have frozen and/or suffocated after hitting space in the TIE. In the Rebels series, Sabine did the same thing. I've looked at what Wookieepedia states about the TIE as well as the old video game Star Wars "TIE Fighter".  Does the movie or any other work explain how Poe and Finn survived? Android app - can I see interesting questions? On the website the default list of questions displays interesting tab (which I believe has questions based on my highest scoring tabs). That's very convenient and allows me to check new questions that are relevant for me easily.   Can the same view be added to the mobile app? Shuffle an array with python, randomize array item order with python What's the easiest way to shuffle an array with python? How to batch resize an image and create multiple sizes (with Photoshop)? I want to batch resize an image in multiple sizes with Photoshop.  The sizes are most of the time the same so I would like to have profiles for a set of fixed sizes. Ideally I just need to open my source image and click on "Batch Profile #1" and then Photoshop is creating/saving X different sizes of that image as an output in a specific folder.  This is what i want: set fontsize for TikZ figure Possible Duplicates:             Voting on questions played a big role in the site when we didn't have the option to comment on, close, migrate, flag, or edit questions which did not fit the site or required modification to be useful and usable.  As it is, question votes are not much more than a popularity contest, and a lot of bandwidth is taken up on Meta regarding question votes.  Notably:        In most cases a downvote is being given where a good edit should be used instead, or a comment indicating that the question user needs to provide more information.  People are also downvoting in the case of duplicates (or suspected duplicates) when they don't want to do the work of finding the duplicate.  There is no need to downvote to 'alert' other users that a question isn't worth spending time on - if it's bad enough that it can't be fixed, and so bad that you want to be sure others avoid it then it's not worth keeping open.  These are all situations where the person downvoting should be taking a different action.  , and several questions have thrown ideas at how to increase question upvoting artificially, and a  was even implemented to encourage this behavior.  Given all this, I am now convinced that voting on questions is an unnecessary distraction.  Voting no longer provides any useful information about a question, and there are more than adequate means of fixing or removing questions that need work.  Voting makes more sense for sites like Meta where questions aren't really questions, but suggestions and discussions.  For purely Q/A sites such as SO, SF, SU, the voting is superfluous.  The only two possible positive things voting now provides are:   A way for users at lower rep levels to have a voice when they are unable to perform many of the other functions that are needed to fix a question (I contend this is useless, but perhaps they need some sort of outlet) A 'hook' to get new users into the reputation game (which fails frequently because questions aren't getting upvotes, especially questions from new users)   If that is truly the case, then we should consider removing question voting altogether, or removing the effect of question votes on reputation.   Am I wrong? What obvious and explicitly important role does question voting play in our community? What Is Energy? Where did it come from? The simplistic undergrad explanation aside, I've never really understood what energy really is. I've been told that it's something when converted from one kind of something to another kind, or does some "work", as defined by us, but what is that something?  Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learned from thermodynamics where it goes, but where does it come from?  I know this sounds like something trivially simple, but there is so much going on in the physical universe and I just can't grasp what it is. Maybe it is because I lack the mathematical understanding that I can't grasp the subtle things the universe is doing. Still, I want to understand what it is doing.  How do I get to the point of understanding what it's doing?  (Note: What prompted me to ask this was . I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.) Who were the other two heroes with Superman in the newspaper article who left? In Justice League, we see a newspaper article showing three heroes who died (left to their home planets) but it was too fast that I missed catching the faces.   Who were the other two heroes who are dead? What is the preferred way to mention Stack Overflow from Twitter? I wanted to tweet something on the lines of "Stack&nbsp;Overflow is awesome". When I do this kind of stuff, I usually use a username (for example: ) or a hashtag ()  I found , but it seems it is from someone unrelated with Stack&nbsp;Overflow.  Is there a Twitter account I can reference, or should I use a hash-tag? Should I just point to the site?  Thanks!  (Notice: I've seen lots of questions regarding Twitter and Stack&nbsp;Overflow, but those are mostly about integration/login with Twitter for the users accounts; I'm just asking for a single account for Stack&nbsp;Overflow; probably for announcements and PR, which is different) My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. Now I want to install Windows 8 for dual boot again, but I have no DVD and do download the ISO one needs a product key. That key is not on the back of the laptop anymore but somewhere on the mainboard.  Is there any way to recover the product key from the mainboard using Ubuntu? How do I programmatically force an onchange event on an input? How do I programmatically force an onchange event on an input?  I've tried something like this:  var code = ele.getAttribute('onchange'); eval(code);   But my end goal is to fire any listener functions, and that doesn't seem to work.  Neither does just updating the 'value' attribute. How to install packages in ubuntu without internet connection? When you reach 10k rep, you get to see other users deleted answers. I just think it's dishonest as a system to call a feature "delete," but show that to totally random strangers in the backroom. Due to the nature of the rep, given enough time, all of the regular users would eventually reach 10k rep, no matter how insensitive or jerk they are.  There's usually a good reason why people hit "delete" to their own work. They regret writing something totally bullshit or maybe they spoke too much about their own work that they shouldn't have.  First of all, is there really a practical reason why 10k rep user should view self-deleted answers? Does the benefit really outweigh the creepiness? (Imagine if total strangers could see your deleted messages in Gmail?)  What suggestions could we make to change it? Borrowing from the email metaphor, I think it makes sense for users to "empty trash" so the items gets seriously deleted. Imho, however the deletion is implemented is not important. It could be a soft delete, SQL Delete, or physical destruction of metal plates as long as the user or the 10k rep user no longer sees the deleted item.  What are the workarounds, if we can't change the behavior?  The least we could do is stop calling it "delete" if it's actually not a "delete." Call it "hide from public" or "actually not delete" or something.  Related user voice:        Edit: Recovery of a deleted-but-good answer is a practical benefit for the community, but I see that as creepy dangerous feature. IMHO, each user should have the rights to take back his or her perfectly good answer, and that the protecting privacy outweighs the benefit or creepiness.   I understand that it's all possible to find things out from Google cache or wherever, but why promote this as default behavior? As 10k rep user, I see others deleted answers by default. @codinghorror wrote:     But like any version control system, deletions are illusory.   Anything and everything that's implemented by human is more or less illusory including phone system and the medical records. Should the government start tapping into phone lines and scanning for words? Remember, this is a question of should had we given the opportunity to design a system. I'm having this Error when I try to compile my .tex file with xelatex:  ! Package hyperref Error: Wrong DVI mode driver option `dvips', (hyperref)                because XeTeX is running.  See the hyperref package documentation for explanation.   This is my LaTeX header  \documentclass[10pt, a4paper, dvips,]{moderncv}  \moderncvtheme[grey, roman]{classic}  \usepackage{ifxetex} \ifxetex % only for xelatex \usepackage{xltxtra} % this should load both fontspec &amp; xunicode \usepackage{fontspec} \setromanfont{Helvetica Neue} \else % \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \fi % \ifxetex   Any ideas?  EDIT:  I removed the 'dvips', now I get:  ! Package hyperref Error: Wrong driver option `pdftex', (hyperref)                because pdfTeX in PDF mode is not detected.  See the hyperref package documentation for explanation.   Now my first latex line  \documentclass[10pt, a4paper]{moderncv}   Any clues ? Why a backslash displays as "W"+strikethrough in Netbeans? I'm facing a strange problem with Netbeans. In the editor, when I type a backslash(), I see a W with a strikethrough (as shown in the attached image).  So the c:\\abc is displaying as:    But, when I copy and paste that string into notepad, it gets pasted normally (i.e. c:\\abc).  Has anyone faced this problem before? How to fix it? What if residuals are normally distributed, but y is not? When closing a question as duplicate the &quot;Possible duplicate of&quot; comment is automagically added to the question. However when adding such a comment manually: &quot;possible duplicate of random text&quot; that comment is also removed. To me it doesn't make sense that questions which to not point to a SO / SE question or even comments without any links in it are also removed when the question is closed as a dupe. I just found this out by close voting a question as a dupe of another question, but I also added a manual comment which linked to a user profile (because it contains lots of relevant information for OP and possible future reasons): possible duplicate of this [user](http://stackoverflow.com/users/727208/teresko?tab=answers&amp;sort=votes)'s list.  By the looks of it the script simply checks whether the comment starts with &quot;possible duplicate of&quot; and if it matches it will be removed. The  team also did a quick test with a very simple example which also got thrown away: possible duplicate of testing........  I understand that this issue is minor and there might as well be a technical reason for this (i.e. not having to keep track of automatic vs manual comments), but this behavior is kinda odd to me. I can think of two solutions for this:  keep track of manually vs automagically added dupe comments (again you SO people might have a good reason to not want this). Improve the &quot;dupe comment matcher ™&quot; to at least check whether the comment is in the correct format for &quot;auto dupe comments ™&quot; with a proper link to a question page. Cannot turn script back on I've got an issue. When (Input.GetMouseButtonUp (0), Speedy script is disabled, thus we cannot enable it again from inside the script like shown below. How can we go around this?  public class  Speedy : MonoBehaviour {  GameObject car;   if (speed == good)     {         GameObject car = GameObject.FindWithTag ("Car");          if (Input.GetMouseButtonUp (0))          {             car.GetComponent&lt;Speedy&gt;().enabled = false;         }          else car.GetComponent&lt;Speedy&gt;().enabled = true;     } }