During the last debate, Gandalf says:     "Concerning this thing, my lords, you now all know enough for the    understanding of our plight, and of Sauron's. If he regains it, your valour    is vain, and his victory will be swift and complete: so complete that none    can foresee the end of it while this world lasts. If it is destroyed, then    he will fall; and his fall will be so low that none can foresee his arising    ever again. For he will lose the best part of the strength that was native    to him in his beginning, and all that was made or begun with that power will    crumble, and he will be maimed for ever, becoming a mere spirit of malice    that gnaws itself in the shadows, but cannot again grow or take shape. And    so a great evil of this world will be removed."   How does Gandalf -or any other of the wise- know that Sauron will indeed be crippled to such an extent if the ring is destroyed (and why is this not already mentioned during the council of Elrond)?    To understand where this question comes from, it may be useful to know that Peter Jackson's depiction of Sauron's incapacitation at the end of the Second Age is controversial. It is not said in the books that a fit Sauron has his ring-finger cut of and then suddenly disintegrates. The text rather suggests that Gil-Galad and Elendil maim Sauron, whereafter Isildur cuts the ring off a near-incapacitated Sauron.    It was brought to my attention that this question may superficially be viewed as a duplicate of . However, that question does not ask after the method by which the wise determine that Sauron will be crippled by the ring's destruction. That is, it asks whether, not how. I find this important because during the last debate Gandalf urges the leadership of Gondor and Rohan basically to bet everything on this single assumption. So one would think/expect that Gandalf has solid proof for his assertion.
Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?  Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of?
Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice) In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?   I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$. Single Variable Calculus Reference Recommendations Sed - unknown option to `s' Boliviana de Aviacion (BOA) tariff codes I am trying to book one-way tickets on BOA on Jan 13 2015 from La Paz (LBP) to Buenos Aires (EZE) from their website.  Their website asks me to make a choice between 7(!) different tariff codes - they are SSEEOB, QEEOB, TMEEOB, VEEOB, HEEOB, BEEOB, YOW. The prices are different for every tariff, but other than that I cannot see any obvious difference and there is no explanation available. Also I am unable to find anything useful on their website  Could anyone from Bolivia or familiar with BOA explain what is going on there and what choice am I supposed to make? 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. Show that the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals May I please ask how to show the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals?  (i.e. Show that $(\Bbb{R}[X])^*$ and $\Bbb{R}^{∞}$ are isomorphic where $\Bbb{R}$ means the set of all reals) "going to" vs "will" Consider the surface $S$ in $\mathbb{R}^3$ given by the graph of $f(x, y) = xy^2 − 2y^2 + e^x$ . Calculate the intersection point of the tangent planes to $S$ above $(x, y) = (0, 0), (0, 1)$, and $(0, 2)$.  Can someone guide me through solving this problem? I am confused on where to start because we didn't cover this in class. Thanks so much What are the alternatives for storing client side encrypted data? I want to store client side encrypted data in Drupal 7, so that it is encrypted on the client end, and stored and transmitted in encrypted form. The password is never transmitted or stored on the server. The only modules I have found which do this are  and  but they are for Drupal 6. I need something that is user specific, so that different users can use different keys, rather than a single key or a set of keys used on the server end. Are there any similar modules for Drupal 7? Usage of brackets/parentheses When and how do we use the different kinds of brackets/parentheses&nbsp;— (){}[]&nbsp;— while writing English (not code)? Suppose $X_1,\ldots,X_n$ are a random sample of $U(-\theta,\theta)$ with $\theta&gt;0$. How can find $$ E\left(|X_1|\ \big|\max |X_i| \right) $$ that $1\leq i\leq n$? Does the universe have an edge/boundary/barrier? If yes, what is at the edge? My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they began? ie space itself is curved...or would they hit some invisible barrier? /usr/sbin/grub-mkconfig: 1: /etc/default/grub: If: not found When creating the id attributes for HTML elements, what rules are there for the value? Is there an equivalent of 'which' on the Windows command line? As I sometimes have path problems, where one of my own cmd scripts is hidden (shadowed) by another program (earlier on the path), I would like to be able to find the full path to a program on the Windows command line, given just its name.  Is there an equivalent to the UNIX command 'which'?  On UNIX, which command prints the full path of the given command to easily find and repair these shadowing problems. Producing PDF/X compliant document with XeLaTeX return eats exception I found the following behavior at least weird:  def errors():     try:         ErrorErrorError     finally:         return 10  print errors() # prints: 10 # It should raise: NameError: name 'ErrorErrorError' is not defined   The exception disappears when you use return inside a finally clause. Is that a bug? Is that documented anywhere?  But the real question (and the answer I will mark as correct) is: What is the python developers' reason to allow that odd behavior? I thought of this question right after I posted a tweet about a service upgrading me to a free student account since I am in college. I said "That really helps a broke college student out." I actually paused for a second while I was writing that to decide if I should say     That really helps out a broke college student.   or     That really helps a broke college student out.   Are there any prescriptive rules about splitting phrasal verbs like this? I know this breaks the "don't put a preposition at the end of a sentence" rule, but that "rule" has an exception for phrasal verbs.  To clarify:   Is there any rule that says phrasal verbs can't be split, even if it is just an imposed, prescriptive rule? Does splitting a phrasal verb to put a preposition at the end of a sentence fall under the phrasal verb exception to the prepositions at the end of a sentence rule? In Visual Studio, we've all had "baadf00d", have seen seen "CC" and "CD" when inspecting variables in the debugger in C++ during run-time.  From what I understand, "CC" is in DEBUG mode only to indicate when a memory has been new() or alloc() and unitilialized.  While "CD" represents delete'd or free'd memory.  I've only seen "baadf00d" in RELEASE build (but I may be wrong).  Once in a while, we get into a situation of tacking memory leaks, buffer overflows, etc and these kind of information comes in handy.  Would somebody be kind enough to point out when and in what modes the memory are set to recognizable byte patterns for debugging purpose? I have an old Facebook account that I created, uploaded a couple of photos and simply ignored (i.e. no friends added). Now, I can't remember the email address associated with the account and I don't remember the password either. How can I find out which email address I had associated with that account?  P.S. I have created a new Facebook account and my picture can be used to identify me (if I can get someone at Facebook to take a look). Other than that, I'm willing to have that old account deleted altogether if recovery isn't possible.  Is there a way to resolve this issue?