In Spivak Calculus, chapter 1 question 11 vi. asks the reader to find all numbers $x$ for which $\lvert x-1\rvert+\lvert x+1 \rvert &lt;1$. Intuitively speaking, it is quite obvious that there is no number $x$ that would make this inequality true, and in checking a graph of the equation I am now certain of this. Despite the textbook not asking for a proof, I am curious as to how one would go about proving this rigorously? Would this require using multiple cases or is it possible without?  Thanks for any insight.  Note: Ideally any proof presented would not require calculus, as that is beyond the scope of the first chapter of the book, but I'd still be interested in any proof involving calculus as this is purely out of curiosity.
Can you help me with this inequality?  $|x-1| +|x+1| &lt;1$.  I think that it doesn't exist an $x$ that satisfies this inequality, but I'm not really convinced about my answer.
Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if:  1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive,  2) $\mathcal{C}$ has a zero object,  3) $\mathcal{C}$ has binary products,  4) $\mathcal{C}$ has kernels and cokernels.  A category $\mathcal{C}$ is abelian if it is preabelian and satisfies:  5) every monomorphism is a kernel and every epimorphism is a cokernel.  Define the coimage of a map to be the cokernel of its kernel, and the image to be the kernel of its cokernel. We have the following commutative diagram:    where $\overline{f}$ is the only existing map (because of universality of kernel and cokernel).  I'm having trouble proving the following:     A preabelian category $\mathcal{C}$ is abelian iff $\overline{f}$ is an isomorphism.   The converse is easily shown, I'm having trouble proving $\Rightarrow$... Are users notified when their questions are reopened? Do users get notified when their question is reopened? I know there are no notifications , and that makes sense because 1) you assume the user is paying attention to their open, unanswered question, and they'll probably notice it's been put on hold and 2) it follows with the SE policy of (in general) not sending notifications for negative events.  However a reopening is a good thing, and once a question is closed it becomes less likely that the OP will be coming back to check on it often (so they might not notice the reopening otherwise). If there are not reopen notifications, why not? Or can we add them? Timeout on a function call When playing as a spellcaster, is it valid the use of the Ready action to prepare Dispel Magic to counter a spell cast by an opponent? In the same vein, when declaring said action, do I have to specify which spell slot I will use when Readying the Dispel Magic spell? If that is the case, does this tactic nullify an opponent's spell? I get the impression that this is basically turning Dispel Magic into a spell with a casting time of &quot;1 reaction&quot;. Is my reading of the rules accurate? Changing Custom Post Type Name Order Does there exist a symmetric matrix $A$  such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \cdot2^{\sqrt{n}}$ for all $n$? I think no. The trace of $A^n$ equals $\sum\limits_{i=1}^n\lambda_i^n$ where $\lambda_i$ are the eigenvalues of $A$. Now, if the absolute value of trace of $A$ is bounded below by $2$, then I think the trace of $A^n$ will grow infinitely. Am I right? Thanks beforehand. How can I edit my network profile independently? My oldest account on the network is Area 51, followed by Mi Yodeya.  I recently filled out the "about me" section on Mi Yodeya (but nowhere else) and did not choose the "update all sites" option because my blurb is pretty specific to that site.  I intended to have that blurb on exactly one site.  I was surprised tonight to discover that that blurb is now on my network profile, where I see no "edit" link.  Am I stuck with either having a site-specific blurb as my network blurb (because it's my first non-Area-51 site) or rewriting it to be more general?  Or is there some way that I can independently edit the network profile?  (Or designate a different site as the default?)   shows that someone else had this problem too, but I'm asking for support, not for a new feature, so I'm asking this separately. Is doing two PhDs a good path? I am studying a combined bachelor of engineering (electrical) and bachelor of mathematics; I just started this year and will graduate in 2018. The reason why I am doing double degrees and not a single degree is because I love both electrical engineering and mathematics and I could not ignore any of them. So with this in mind, I am thinking of doing two PhDs when I graduate (one in electrical engineering and one in mathematics). Is this a good path or I should concentrate on only one of them? Should an off-topic question always be downvoted? The upvote tooltip says:      This question shows research effort; it is useful and clear.   The downvote tooltip says:      This question does not show any research effort; it is unclear or not useful.   Sometimes, a question can show research effort, be useful, be clear, but just not in the good place. Voting to close the question and downvoting seems like a double penalty.  Should an off-topic question always be downvoted ? We are provided with a string A (may contain repeated characters as well ). One needs to make all possible permutations of A , and find out how many pairs of strings chosen from these permutations are not similar.  Two strings are similar if we can make both of them exactly same by carrying out at most 1 swap in each of them .  Example :- if we are provided with string abcd  , then for permutation abcd , the permutations which are not similar to it are bcda , bdac , cadb , cdba , dabc , dcab . Similar is the case for other permuations as well. So overall number of desired pairs are 24*6=144 .  I tried solving it for each possible pair by approach mentioned here     But as the permutations can be large enough , I need a better approach of solving this. If $g\circ f$ is injective and $f$ is surjective then $g$ is injective Definition of "pepper" in hash functions In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian Let $G$ be a group. If for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then phow can I prove that  $G$ is abelian? Downgrade nVidia Shield to Android 4.4 KitKat Let $M_n$ denote the vector space of all $n\times n$ real valued matrices and for any matrix $A\in M_n$ let $A^t$ denote its transpose. Define the linear map $T:M_n\rightarrow M_n$ as $$T(A)=A+A^t.$$ What is the minimal polynomial of $T$? What are the eigenvalues and the corresponding eigenspaces of $T$? What is the characteristic polynomial of $T$?  I learned about minimal polynomials of operators from $\textit{Linear Algebra Done Right}$ by Axler. In his description he explains how in order to find the minimal polynomial you must find the smallest positive integer $m$ such that $T^m$ is a linear combination of $I, T, T^2, \ldots, T^{m-1}$. That is, the smallest integer $m$ such that $$T^m = a_0I+a_1T+a_2T^2+ \cdots +a_{m-1}T^{m-1}.$$  As for the eigenvalues...since we are working with real-valued matrices, then $A$ and $A^t$ have the same eigenvalues, so if $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ (and hence of $A^t$) wouldn't $2\lambda$ be an eigenvalue of $A+A^t$? I apologize if I am saying utter nonsense... Show that the boundary of a set equals the boundary of its complement $\newcommand{\bdy}{\operatorname{bdy}}$  I'm trying to show that $\bdy(A) = \bdy(A^c)$.  I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know where to go from there.  Any help or hints would be very much appreciated. How to renew an expired keypair with gpg What is the best way to renew a gpg key pair when it got expired and what is the reason for the method?  The key pair is already signed by many users and available on public servers.   Should the new key be a subkey of the expired private key?  Should it be signed by the old (I could try to edit the key and change the date of expiration to tomorrow)? Should the new key sign the old? Why did Voldemort stop employing Occlumency against Harry in Deathly Hallows? I've recently installed Ubuntu on Oracle's Virtualbox.  Now I want to install the guest additions.  I started up my Ubuntu guest OS, clicked "Devices" and chose "Install Guest Additions". Nothing happened, and I don't see the CD icon on my desktop.  How can I make this work? Prepositions used with "inquire"